Studies 
in  spher- 
ical and 
practical 
astronomy 


GIFT   OF 
Mrs*  W..  W.   Campbell 


ASTRONOMY  DEPT» 


BULLETIN    OF  THE   UNIVERSITY  OF   WISCONSIN 

SCIENCE  SERIES,  VOL.  1,  No.  3,  PP.  57—107. 


STUDIES  IN  SPHERICAL  AND  PRACTICAL 
ASTRONOMY 


BY 


^         STUOKNTS* 

A    OBSERVATORY 
GEORGE  C.  QOMSTOCK  V^i 

Director  of  the  Washburn  Observatory\^J/'+ 

"OF 


PUBLISHED   BY  AUTHORITY  OF  LAW  AND  WITH  THE  APPROVAL  OF 
THE  REGENTS   OF  THE   UNIVERSITY 


MADISON,  WIS. 

PUBLISHED  BY  THE  UNIVERSITY 
JUNE,  1895 


PRICE  40  CENTS 


of 


CHARLES  KENDALL  ADAMS,  PRESIDENT  OF  THE  UNIVERSITY 

EDITORS 

GEORGE  L.  HENDRICKSON,  Philology  and  Literaturt 

WILLIAM  H.  HOBBS  (Chairman),  Science 

DUGALD  C.  JACKSON,  Engineering 

FREDERICK  J.  TURNER,  Economics,  Political  Scitnce,  and  History 

ASTRONOMY  DEPTi 


*• " 


Democrat  Printing  Company,  State  Printer 


Studies  in  Spherical  and  Practical  Astronomy  -  Comstock 


Minor  suggestions  -  Page 

The  reduction  of  level  readings,  57 

To  focus  a  telescope.  58 

I.  A  simple  but  accurate  expression  for  the      60 
atmospheric  refraction. 

II.  To  correct  the  sun's  declination  for  the      64 
effect  of  refraction. 

III.  Determination  of  the  angular  equivalent  of     68 
one  division  of  a  spirit  level. 

IV.  The  simultaneous  determination  of  flexure,     75 
inequality  of  pivots,  and  value  of  a  level 
division  for  a  "broken"  transit. 

V.  Determination  of  time  and  azimuth  from        81 
transits  over  the  vertical  of  the  pole  star. 

VI.  Determination  of  latitude  and  time  from       94 
equal  altitudes  of  stars. 


M177041 


BULLETIN    OF  THE   UNIVERSITY   OF  WISCONSIN 
SCIENCE  SERIES,  VOL.  1,  No.  3,  PP.  57-107,  JUNE,  1895 


STUDIES  IN  SPHERICAL  AND  PRACTICAL 
ASTRONOMY. 

BY  GEORGE  C.  COMSTOCK, 
Director  of  the  Washburn  Observatory. 

The  following  pages  contain  an  exposition  of  methods 
for  the  treatment  of  certain  problems  in  spherical  and  prac- 
tical astronomy,  which,  from  his  own  experience,  the 
author  has  found  to  be  advantageous  in  practice.  For  the 
most  part  these  methods  are  original  and  hitherto  unpub- 
lished, but  in  part  they  are  due  to  others,  whose  published 
exposition  of  them  is  not  readily  accessible  to  American 
students.  In  cases  of  the  latter  kind  due  acknowledgement 
is  made  in  connection  with  the  presentation  of  the  subject 
matter,  but  I  have  not  scrupled  to  modify  or  to  completely 
alter  the  mode  of  presentation  of  those  subjects  which 
have  been  treated  by  others,  adopting  in  each  case  that 
method  which  has  seemed  to  me  simplest  and  most  easily 
followed. 

MINOR  SUGGESTltfOS. 

The  Reduction  of  Level  Readings. — To  determine  the  incli- 
nation of  a  nearly  horizontal  line  or  plane  by  use  of  a  spirit 
level,  Chauvenet '  gives  rules  which  in  all  cases  require  the 
same  operations  to  be  performed  with  the  level,  but  in 
which  the  mode  of  treatment  of  the  level  readings  depends 
upon  the  manner  in  which  the  scale  is  graduated,  one 
method  when  the  zero  is  at  the  end  of  the  scale  and  another 
when  it  is  in  the  middle  of  the  scale.  The  modes  of  re- 
duction are  sufficiently  Illustrated  in  the  following  ex- 
amples given  by  Chauvenet.2 

i  Spherical  and  Practical  Astronomy,  Vol.  II,  §§  52,  55. 
*Loc.cit. 


58  BULLETIN    OP    THE    UNIVERSITY    OP   WISCONSIN 

Zero  at  end.  Zero  in  middle. 
W.          E.  W.  E. 

29.1        31.2  +64.0        +13.5 

35.4        24.9  -  10.1        -  60.7 


64.5        56.1  +77.5 

56.1  -70.8 


z  —  8.4  -4-  4  =  2.1  div.  z=  +  6.7  -=-  4  =  +  1.675  div. 

A  method  of  reduction  which  is  the  same  for  both  types 
of  level,  and  which  is  in  most  cases  more  convenient  than 
the  above,  is  as  follows:  In  the  square  array  of  numbers 
which  constitute  the  observed  readings  of  the  level,  take 
the  diagonal  differences.  The  mean  of  the  two  diagonal 
differences  is  the  inclination  of  the  line  in  half  divisions  of 
the  level.  That  end  of  the  line  is  the  higher  which  is  ad- 
jacent to  the  greatest  single  reading.  If  the  level  readings 
have  been  correctly  made  the  two  diagonal  differences  will 
be  the  same,  and  the  reduction  thus  serves  as  a  check  upon 
the  accuracy  of  the  record. 

Thus,  from  the  readings  given  above,  we  see  at  a  glance 
that  in  the  first  case  z  =  4.2  half  divisions  and  the  readings 
have  been  correctly  made.  In  the  second  case  2  =  3.35  half 
divisions  and  there  is  a  discrepancy  of  0.1  div.  in  the 
readings. 

Although  I  cannot  doubt  that  this  simple  mode  of  reduc- 
ing level  readings  has  been  employed  by  others,  I  do  not 
find  it  in  any  of  the  treatises  upon  practical  astronomy  to 
which  I  have  access. 

To  Focus  a  Telescope. — Let  the  telescope  be  directed  to  a 
circum-polar  star  near  culmination  and  introduce  between 
the  objective  and  the  star  an  opaque  screen  pierced  with  a 
circular  aperture  from  10  to  20  mm  in  diameter.  As  the 
aperture  is  moved  about  in  front  of  the  objective  an  image 
of  the  star  will  be  formed  by  different  parts  of  the  objec- 
tive, and  if  the  telescope  is  not  perfectly  focused  these  im- 
ages will  fall  at  slightly  different  parts  of  the  field;  e.  g., 
let  the  aperture  be  held  opposite  the  upper  part  of  the 
objective  and  the  star's  image  be  bisected  with  a  horizontal 


COMSTOCK — STUDIES    IN    ASTRONOMY  59 

thread.  Then  shift  the  aperture  to  the  lowest  part  of  the 
objective  and  note  whether  the  image  of  the  star  is  sensibly 
displaced  from  the  thread.  If  the  image  moves  in  the  same 
direction  with  the  aperture  in  the  screen,  the  eye  end 
should  be  drawn  out;  if  in  the  opposite  direction  it  should 
be  pushed  in  until  a  position  is  found  at  which  there  is  no 
displacement  of  the  star  image. 

By  this  process  the  telescope  may  be  so  adjusted  that 
the  error  of  focusing  shall  not  exceed  1 : 10000  part  of  the 
focal   length,  provided  it  is   so  firmly  supported   as  to  be 
free  from  the  effect  of  accidental  tremors  and  vibrations, 
e.  g.  the  telescope  of  a  transit  instrument. 


60  BULLETIN   OF   THE    UNIVERSITY   OF   WISCONSIN 


I.-A  SIMPLE  BUT  ACCURATE  EXPRESSION  FOR  THE 
ATMOSPHERIC  REFRACTION. 

Bessel's  expression  for  the  refraction  ' 

R  =  a/3A  y^tanz 

is  commonly  employed  for  all  accurate  computations  of  the 
refraction,  and  when  so  employed  requires  that  the  five 
quantities,  a,  p,  y,  A,  A,  shall  be  interpolated  from  specially 
prepared  refraction  tables.  It  is  the  purpose  of  the  present 
paper  to  so  transform  this  expression  that  the  refraction 
may  be  computed  without  recourse  to  these  tables. 

Since  the  refraction  admits  of  development  in  terms  of 
the  odd  powers  of  tanz,  we  may  write  for  the  mean  re- 
fraction : 

JBm  =  a  tan  z  =  a^  tan  z  —  as  tan*  z etc. 

=  a^  f i  — '  JIT  tan*zj  tan  z     (approximately 

The  Pulkowa  Refraction  Tables  are  presumably  the  most 
accurate  ones  available  at  the  present  time,  and  from  these 

tables  I  find: 

»  » 

al  =  57.584  as  =  0.0640 

If  with  these  values  we  compute 


and  compare  it  with  the  tabular  values  of  a  we  shall  find 
the  following  satisfactory  agreement: 

z                   0°              20°             40*              60'  75* 

•                            •                           f                            I  It 

Tabular  a        57.586        57.577        57.538        57.386  56.694 

Formula          57.584        57.576        57.537        57.391  56.693 

The  quantity  A  is  a  complicated  function  of  the  zenith 

:  Tab.  Rtg.,  LXH. 


COMSTOCK — STUDIES  IN  ASTRONOMY  61 

distance,  z,  but  for  values  of  z  less  than  75°  it  may  be  rep- 
resented by  the  empirical  formula: 

A  =  1  +  h  tan*z  h  =  0.001362 

The  following  comparison  shows  the  degree  of  accuracy 
with  which  this  formula  represents  the  tabular  numbers : 

z  50°  60°  70°  75° 

Tabular  A  1.0022  1.0044  1.0103  1.0188 

Formula  1.0019  1.0040  1.0103  1.0190 

If  we  represent  by  e  the  adopted  coefficient  of  expansion 
of  air  per  degree  C.,  by  r0  the  normal  temperature  of  the 
refraction  tables,  and  by  r  any  other  temperature,  we  shall 
have : ' 

,        r  I"* 

r*  =   [l  +  £  (r  -  r0)  J 

Developing  this  expression  by  means  of  the  exponential 
series  it  becomes,  when  the  terms  of  the  order  s*  are  neg- 
lected, 

-j-  E  (r  —  r0  \  h  tan*z 


1  —  E  h  tan2z  (T  —  r( 


For  zenith  distances  less  than  75°  the  exponent  A  does 
not  sensibly  differ  from  unity,  and  we  have 


where  B0  is  the  normal  barometric  pressure  of  the  tables 
and  B  is  the  actual  pressure  at  any  time,  i.  e.  the  reading 
of  the  barometer  "reduced  to  the  freezing  point." 

Collecting  the  expressions  for  the  several  factors  above 
developed,  we  obtain: 

r 

R  =  #!  -g-  £_1]jrro  tan  z  \     1  —    f^  +  £  h  (r  —  rj\tan*z 
i  Chauvenet,  Vol.  II,  p.  165. 


62 


BULLETIN  OF  THE  UNIVERSITY  OF  WISCONSIN 


From  the  Pulkowa  Tables  we  find: 

o  o 

£0  =  751.5  mm.  r0=9.31C.  £-*  =  271.05  C. 

Denoting  the  quantity  enclosed  in  brackets  by  F  and  ii 
troducing  numerical  values,  we  obtain: 

BF 


It  =  [l.33207] 


tan  z 


271,05  +  r 
log  F  =  -  (46.2  +  0.22  r)  tan*z 

In  the  use  of  these  formulae  B  and  T  must  be  expresse 
in  millimeters  and  degrees  C.  The  formula  gives  log  F  i 
units  of  the  fifth  decimal  place.  The  number  enclosed  i 
brackets  is  a  logarithm. 

The  corresponding  formulae,  when  the  pressures  are  e: 
pressed  in  English  inches  and  the  temperatures  in  degree 
F.,  are: 

R  =  [2.992151  AKK£nF. —  tan  z 

L  J  455.9  +  r  g 

log  F  —  —  (42.3  +  0.12  r)  tan*z 

The  computation  by  these  formulae  is  not  more  laborioi 
than  the  direct  computation  from  the  tables,  and  the  fo 
lowing  comparison  shows  that  the  differences  between  tl 
formulae  and  the  tables  are  far  less  than  the  uncertainty  i 
the  tabular  numbers  themselves.  For  zenith  distances  nc 
much  exceeding  75°  the  formulae  may  be  considered  fc 
most  purposes  a  complete  equivalent  for  the  tables : 

COMPARISON  OF  THE  REFRACTIONS  FURNISHED  BY  THE 
FORMULAE  AND  BY  THE  PULKOWA  TABLES. 


Barometer  

765.0mm 

28.500m 

765.0mm 

28.500m 

Att.  Thermom  . 

O.OC 

70.  OF 

0.0  C 

70.  OF 

Ext.  Thermom. 

-25.  OC 

75.  OF 

-25.0C 

75.  OF 

75° 

75° 

60° 

60° 

Tabular  Ref... 

w 

246.02 

192.83 

g 

115.36 

90.65 

Formula  A  

246.03 

192.84 

115.36 

90.66 

Formula  B  

246.02 

192.84 

115.35 

90.66 

COMSTOCK — STUDIES  IN  ASTRONOMY  63 

The  coefficients  in  equations  A  and  B  have  been  so  de- 
termined as  to  reproduce  with  all  possible  fidelity  the  re- 
fractions of  the  Pulkowa  Tables,  but  they  may  be  made  to 
represent  the  actual  refractions  with  greater  precision  by 
the  application  to  the  constant  coefficients  of  the  formulae 
of  certain  corrections  depending  upon  the  latitude  of  the 
place  at  which  the  refraction  is  required,  the  amount  of 
moisture  in  the  air  and  the  wave  length  of  the  light  whose 
refraction  is  to  be  computed.  These  corrections  are  de- 
veloped in  Vol.  IX,  Publications  of  the  Washburn  Observa- 
tory. The  most  important  of  them,  and  the  only  one  which 
need  be  considered  here,  is  that  depending  upon  the  lati- 
tude. Its  effect  will  be  sufficiently  taken  into  account  by 
adding  to  the  bracketed  coefficient  in  the  equations  A  and 
B,  the  quantity 

C  =  225  sin  (<p  -  60°)  sin  (<p  +  60°) 

where  9  denotes  the  latitude  and  C  is  given  in  units  of  the 
fifth  decimal  place. 


64  BULLETIN  OF  THE  UNIVERSITY  OF  WISCONSIN 


II.—  TO  CORRECT  THE  SUN'S  DECLINATION  FOR  THE  EFFECT 
OF  REFRACTION. 

A  useful  application  of  the  formulae  of  the  preceding 
section  occurs  in  connection  with  the  use  of  the  solar  com- 
pass. It  is  here  required  to  set  off  upon  a  certain  divided 
arc  the  apparent  declination  of  the  sum,  i.  e.  the  true  de- 
clination corrected  for  the  effect  of  refraction.  This  cor- 
rection is  usually  interpolated  from  rather  cumbrous  tables 
of  double  entry.1 

Denoting  the  refraction  in  declination  by  d  and  represent- 
ing by  q  the  parallactic  angle  of  the  sun,  we  have: 


_  _          r>   -jp 

d  =  JR  cos  q  =     2.99215     -r^-   —  tan  z  cos  q 
L  J  4oo  -{-  r 


(I) 


By  applying  the  fundamental  formulae  of  spherical  trig- 
onometry to  the  spherical  triangle,  Pole  —  Zenith  —  Sun, 
and  differentiating  the  equations,  we  find  : 

dA 

——  =  cos  o  cos  q  cosec  z  (2) 

Eliminating  cos  q  between  these  equations,  we  obtain 

d  =  [2.992151  ,.f  F    sec  d  sin  z  tan  z  ^  (3) 

L  J  4oo  -\-  r  at 

where  z,  A,  s  and  t  represent  respectively  the  zenith  dis- 
tance, azimuth,  decimation,  and  hour  angle  of  the  sun. 

dA 
The  numerical  value  of  ~^-  varies  with  the  position  of 

the  sun  in  the  heavens,  but  may  be  readily  determined  at 
any  time  as  follows  :  Let  the  horizontal  circle  of  the  solar 
compass  or  transit  be  set  to  read  some  integral  10'  and  the 
telescope  be  than  pointed  upon  the  sun  by  rotating  the  in- 
strument about  the  lower  motion.  The  sun  having  been 
brought  into  the  field  of  view,  the  earth's  diurnal  motion 

See  Johnson's  Theory  and  Practice  of  Surveying,  pp.  47,  48. 


COMSTOCK STUDIES  IN  ASTRONOMY  65 

will  carry  the  sun  across  the  vertical  thread  of  the  instru- 
ment, and  the  time  at  which  one  edge  of  the  sun  is  just' 
tangent  to  the  thread  should  be  noted  to  the  nearest  sec- 
ond upon  a  watch.  Let  the  instrument  be  now  turned 
upon  the  upper  motion,  keeping  the  lower  motion  clamped, 
in  the  direction  of  the  sun's  movement,  and  the  vernier  set 
at  the  next  integral  10'.  The  time  at  which  the  sun's  edge 
again  becomes  tangent  to  the  vertical  thread  should  be 
noted  as  before.  If  we  represent  by  n  the  interval,  in 
seconds,  between  the  two  observed  times,  we  shall  have : 

dA  _40 
dt    ~'~  n 

If  desired,  the  transit  may  be  set  so  that  the  second 
vernier  reading  is  20',  30',  etc. ,  greater  than  the  first  read- 
ing, and  we  shall  then  have: 

dA         80         120 

— jj  =  —  =  — etc.  and 

dt         nz         ns 

n  =  \  nz  =  £  n8 . . . . etc. 

This  value  of  the  differential  coefficient  enables  us  to 
express  equation  (3)  in  a  form  adapted  to  field  use,  but 
since  for  this  purpose  an  error  of  even  several  seconds  in 
the  value  of  d  is  of  small  consequence,  we  shall  introduce 
some  modifications  in  the  formula  which  will  render  it 
more  convenient  without  seriously  impairing  its  accuracy. 

o 

The  declination  of  the  sun  can  never  exceed  23.5,  and  we 
therefore  write  in  the  place  of  sec  d  its  mean  value,  1.051. 
We  also  put  in  place  of  the  temperature  r  a  mean  value, 
50°  F.,  and  assume  for  the  barometric  pressure  30  inches 
of  mercury.  With  these  modifications  equation  (3)  be- 
comes : 

,  _  [3.3854] Fain  z  tan  z 
n 

We  may  put  the  numerator  of  this  fraction  equal  to 
100  N  and  tabulate  the  values  of  N  with  the  argument  the 
sun's  altitude,  h  =  90°  —  z,  as  follows: 


66 


BULLETIN  OF  THE  UNIVERSITY  OF  WISCONSIN 


h 

N 

h 

N 

10° 

131* 

30° 

36' 

45 

14 

15 

86 

40 

22 

24 

9 

20 

62 

50 

13 

15 

6 

25 

47 

60 

7 

11 

4 

30 

36 

70 

3 

We  now  helve  for  the  refraction  in  declination : 

d  =  100  — 
n 

The  altitude  of  the  sun,  7i,  should  be  noted  on  the  verti- 
cal circle  of  the  instrument  to  the  nearest  half  degree  at 
the  time  of  determining  n. 

The  tabulated  values  of  N  correspond  to  a  temperature 
of  50°  F.  and  a  barometric  pressure  of  30  inches.  They 
may  be  adapted  to  any  other  temperature  by  diminishing 
d  by  one  per  cent  for  each  5°  by  which  the  temperature 
exceeds  50°,  or  by  increasing  one  per  cent  for  each  5°  be- 
low 50°,  but  this  correction  and  the  correction  for  varia- 
tion of  the  barometer  can  usually  be  neglected.  At  great 
elevations  the  barometric  pressure  becomes  so  much  re- 
duced that  its  variation  must  be  taken  account  of,  and  this 
may  be  done  by  diminishing  d  by  one  per  cent  for  each 
300  feet  of  elevation  above  the  sea. 

The  following  examples  will  serve  to  illustrate  the  ap- 
plication of  the  formulae  above  developed.  On  the  after- 
noon of  May  12,  1894,  at  a  place  in  latitude  43°  5'  N.,  lon- 
gitude approximately  90°  west  of  Greenwich,  I  took  the 
following  observations  with  an  engineer's  transit: 


COMSTOCK — STUDIES    IN    ASTRONOMY 


67 


Vernier. 


170        0 
170      10 


Watch. 


h.  m.  s. 
4  5  30 
4  6  27 


Vertical  Circle  =  32°  8' 


d  =  58" 


Vernier. 


170      10 
170      20 


Watch. 


h.  m.  s. 
4  23  18 
4  24  17 


Vertical  Circle  =  28°  46' 


^=39 


By  a  direct  computation  from  the  formula1 

d  =  57"  cot  (S  +  N) 

where  N  denotes  the  Bessel  auxiliary,  I  find  for  the  refrac- 
tion in  decimation  at  the  time  of  these  observations  59" 
and  67"  respectively,  thus  showing  an  agreement  far  within 
the  limits  of  error  permissible  in  surveying  practice. 

If,  as  is  often  the  case,  an  accuracy  of  20"  is  sufficient, 
and  the  altitude  of  the  sun  is  not  less  than  10°,  we  may 
dispense  with  the  tabular  values  of  N  and  write 

d  =  2000  -r-  hn 

where  h  is  the  altitude  in  degrees  and  the  value  of  d  is 
given  in  minutes  of  arc.  The  error  of  this  formula  in  the 
preceding  cases  is  7"  and  4",  respectively. 

i  Chauvenet,  Spherical  and  Practical  Astronomy,  Vol.  I,  p.  171. 


68  BULLETIN  OF  THE  UNIVERSITY  OF  WISCONSIN 


III.— DETERMINATION   OF  THE  ANGULAR   EQUIVALENT   OF 
ONE  DIVISION  OF  A  SPIRIT  LEVEL. 

The  methods  most  in  use  in  this  country  for  the  deter- 
mination of  the  value  of  one  division  of  a  level  require 
that  the  level  should  be  attached  either  to  a  level-trier  or 
to  a  telescope  provided  with  a  good  micrometer.  In  field 
astronomy  it  frequently  happens  that  neither  of  these  aux- 
iliaries is  available  and  the  following  method,  which  in 
respect  of  precision  is  not  inferior  to  either  of  the  others, 
may  be  employed  with  advantage  since  it  requires  no  aux- 
iliary apparatus  other  than  a  theodolite  or  engineer's  tran- 
sit. The  original  suggestion  of  this  method  is  supposed 
to  be  due  to  Braun.1 

Let  the  spirit  level  be  firmly  attached  to  a  theodolite 
which  is  thrown  out  of  a  level  so  that  its  vertical  axis  makes 
an  angle  of  from  1°  to  3°  with  the  true  vertical.  It  is  prac- 
tically convenient  to  so  attach  the  level  that  the  radius  of 
curvature  drawn  through  the  middle  point  of  its  scale  shall 
be  approximately  parallel  to  the  vertical  axis  of  the  theod- 
olite, i.  e.  the  level  shall  be  in  adjustment.  As  the  theod- 
olite is  turned  about  its  vertical  axis  the  level  bubble  will 
run  from  one  end  of  its  tube  to  the  other  and  back  again 
during  a  complete  revolution  of  the  instrument,  and  two 
positions,  two  readings  of  the  azimuth  circle,  may  be  found 
in  which  the  bubble  will  stand  near  the  middle  of  its  scale. 
A  small  turning  of  the  instrument  either  way  from  one  of 
these  positions  will  produce  a  corresponding  small  motion 
of  the  bubble  in  its  tube,  and  this  turning  of  the  theodolite 
and  resulting  motion  of  the  bubble  may  be  made  to  furnish 
not  only  the  value  of  a  division  of  the  level,  but  also  a  test 
of  the  uniformity  of  its  curvature. 

To  determine  the  relation  between   the   readings  of  the 

l  Astronomische  Nachrichten,  No.  2490. 


COMSTOCK — STUDIES  IN  ASTRONOMY 


69 


azimuth  circle  of  the  theodolite  and  the  readings  of   the 
level  bubble  upon  its  scale,  let  the  accompanying  figure 


represent  a  portion  of  the  celestial  sphere  adjacent  to  the 
zenith,  Z,  and  let  V  and  S  be  the  points  in  which  the  axis 
of  the  theodolite,  and  the  line  drawn  from  the  center  of 
curvature  of  the  level  tube  through  the  middle  of  the  bubble, 
respectively,  intersect  the  sphere.  The  arc  SV  is  the  in- 
tersection with  the  celestial  sphere  of  a  plane  passing 
through  S,  V,  and  the  center  of  curvature  of  the  level 
tube,  and  if  the  adjustment  of  the  level  above  referred  to 
is  approximately  made,  VS  may  be  considered  as  the  inter- 
section with  the  sphere  of  the  plane  in  which  the  curva- 
ture of  the  level  tube  lies,  so  that  as  the  bubble  moves  in 
its  tube  its  successive  positions,  when  projected  upon  the 
sphere  will  lie  along  VS,  and  any  position  may  be  identi- 
fied by  its  distance  from  V,  represented  in  the  figure  by  p. 
Since  the  bubble  always  stands  at  the  highest  part  of  the 
tube,  its  position,  S,  and  the  corresponding  value  of  p  are 
found  by  letting  fall  a  perpendicular  from  the  zenith  upon 
tne  arc  VS,  and  in  the  right-angled  spherical  triangle  thus 
formed  we  have  the  relation, 

tan  p  =  tan  y  cos  t 
where  r,  as  it  appears  from   the  figure,   is  the  angle  by 


70  BULLETIN  OF  THE   UNIVERSITY  OP   WISCONSIN 

which  the  axis  of  the  theodolite  is  deflected  from  the  true 
vertical. 

Since  the  level  tube  turns  with  the  theodolite  when  the 
latter  is  revolved  in  azimuth,  while  the  positions  of  the 
points  V  and  Z  remain  unchanged,  it  appears  that  the 
angle  t  must  vary  directly  with  the  readings  of  the  azimuth 
circle,  since  it  measures  the  inclination  of  the  plane  of  the 
level  tube  to  a  fixed  plane  passing  through  the  vertical 
axis  of  the  instrument.  If  we  represent  by  AO  the  reading 
of  the  circle  when  the  arc  VS  is  made  to  coincide  with  VZ, 
we  shall  have  corresponding  to  any  other  reading  A'  : 

tan  p  =  tan  y  cos  (A0  —  A')  (1) 

The  value  of  A0  in  any  given  case  may  be  determined 
by  finding  two  positions  of  the  instrument,  circle  readings 
Ai  and  A2,  in  which  the  bubble  stands  at  the  same  part  of 
the  tube.  Since  the  values  of  p  corresponding  to  these 
two  readings  are  equal,  we  must  have: 

A0-A1=AS-A0        and        A0  =  ^(A1+At) 

If  A'  and  A"  denote  slightly  different  readings  of  the 
azimuth  circle,  and  &'  and  bff  the  corresponding  readings  of 
the  middle  of  the  bubble  on  the  level  scale,  we  may  write 
two  equations  similar  to  equation  (1),  and  taking  their 
difference  obtain: 

sin  (p'  —  p")  A'  —  A"    .    /  .         A'4-A"\.  ,~. 

—  —,  -  —  7  =  2  sin  -  jr  -  sin  I  Aa  --  ^  -  I  tan  y  (2) 

cos  p  cos  p  2  \  2       / 

Since  p'  —  p"  is  the  distance  moved  over  by  the  bubble, 
we  may  write  p'  —  p"=(b'  —  &")$,  where  d  is  the  value  of  a 
division  of  the  level,  and  transform  (2)  into 


2  tany  cos*p  sin  4  (A'  -  A")     sin  L^°  ~  *  (A'  + 
sin  I-  b'  -  b" 

In  this  equation  cosPp  may  usually  be  put  equal  to  1,  or 
its  actual  value  may  be  found  from  the  average  value  of  p 
given  by  equation  (1).  Every  other  factor  in  the  second 
member  of  this  equation  is  known  with  exception  of  tan  y, 
and  the  determination  of  y  will  determine  d. 


COMSTOCK — STUDIES  IN  ASTRONOMY  71 

For  this  purpose  the  instrument  should  be  carefully  lev- 
elled at  the  beginning  of  the  work  and  the  telescope  di- 
rected at  some  object,  approximately  at  right  angles  to  the 
line  joining  two  of  the  leveling  screws  of  the  instrument. 
Let  the  zenith  distance,  »',  of  this  object  be  determined 
from  readings  of  the  vertical  circle  taken  Circle  Right  and 
Circle  Left.  The  vertical  axis  is  now  to  be  deflected 
toward  the  object  by  turning  the  leveling  screws,  and  the 
zenith  distance  of  the  object,  reckoned  from  the  vertical 
axis  of  the  instrument,  z",  is  to  be  determined  from  circle 
readings  in  the  same  manner  as  z'.  We  then  have,  ob- 
viously, 

y  =  z'  -  z" 

To  make  sure  that  the  deflection  of  the  axis  lies  in  the 
plane  passing  through  the  object  sighted  upon,  it  is  well 
to  note  the  position  of  the  bubble  of  that  level  of  the  in- 
strument which  is  at  right  angles  to  the  telescope  tube. 
The  leveling  screws  must  be  so  turned  that  the  reading  of 
the  bubble  of  this  level  on  its  scale  is  approximately  the 
same  after  deflection  as  before. 

By  comparison  with  micrometric  apparatus,  this  deter- 
mination of  y  and  the  resulting  value  of  d  may  seem  crude, 
but  with  a  vertical  circle  reading  to  minutes  only,  the 
values  of  z'  and  z"  can  be  determined  within  30",  and  if  y 
be  made  3°,  d  will  be  determined  with  a  probable  error  of 
one  part  in  four  hundred,  an  accuracy  quite  sufficient  for 
even  the  most  delicate  level.  The  value  of  y  should  be 
between  1°  and  3°,  a  coarse  vertical  circle  and  fine  hori- 
zontal circle  corresponding  to  the  larger  limit,  and  the 
reverse  conditions  to  the  smaller  one. 

To  illustrate  the  method,  I  select  the  following  partial 
investigation  of  the  microscope  level  of  a  small  universal 
instrument,  Bamberg  No.  2598.  The  level  was  investigated 
by  means  of  the  circles  of  the  instrument  to  which  it  was 
attached,  without  removing  or  in  any  way  disturbing  it: 


72 


BULLETIN    OF    THE    UNIVERSITY    OF    WISCONSIN 


DETERMINATION   OF  Y. 

Instrument.  Circle  R.  Circle  L. 

Levelled  ........     180°    26'    49"        358°    4'    13* 

Deflected  .......     179      27       3         359      3     48 

=  0°  59'  40.5* 


z. 

91°    11'    18" 
90      11     37.5 


After  the  level  readings  which  follow  were  completed, 
these  circle  readings  were  repeated  with  the  instrument 
deflected  and  subsequently  leveled,  giving  a  second  deter- 
mination of  y  =  0°  59'  42".  I  adopt  : 

y  =  0°  59'  41' 

The  following  are  the  bubble  observations  in  the  de- 
flected position  of  the  instrument: 


BUBBLE. 

CIRCLE. 

•  BUBBLE. 

BUBBLE. 

CIRCLE. 

BUBBLE. 

25.3-0.8 

111°  34' 

-  0.7    25.6 

26.3      0.2 

291°  6' 

0.4    26.0 

27.7      1.6 

24 

1.8    28.1 

28.1      2.3 

16 

1.9    27.7 

30.2      4.1 

14 

3.9    30.1 

30.2      4.2 

26 

3.9    29.7 

32.6      6.5 

111      4 

6.4    32.6 

32.3      6.4 

36 

6.2    32.1 

34.8      8.6 

110    54 

8.7    35.0 

34.0      8.1 

46 

8.5    34.2 

37.0    10.9 

44 

10.8    37.0 

36.2    10.2 

56 

10.6    36.5 

The  observations  began  with  the  level  bubble  at  one  end 
of  its  scale,  circle  reading  111°  34',  and  the  instrument 
was  turned  through  successive  intervals  of  10'  until  the 
bubble  reached  the  opposite  end,  when  the  settings  were 
repeated  in  the  inverse  order  to  eliminate  the  effect  of  any 
slight  change  in  the  instrument  or  level.  The  instrument 
was  then  turned  into  the  position  corresponding  to  the 
second  set  of  circle  readings  which  were  taken  with  the 
bubble  running  from  one  end  of  the  tube  to  the  other, 
in  both  directions. 

The  mean  of  the  four  readings  of  the  ends  of  the  bubble 
corresponding  to  any  circle  reading  may  be  adopted  as  the 


COMSTOCK — STUDIES  IN  ASTRONOMY 


73 


corresponding  reading  of  the  middle  of  the  bubble,  and 
these  mean  readings  are  given  in  the  following  table: 


CIRCLE. 

BUBBLE. 

CIRCLE. 

BUBBLE. 

T 

26 

DlFF. 

111°34' 

12.35 

291°  6' 

13.22 

89°46' 

25.57 

4.23 

24 

14.80 

16 

15.00 

89  56 

29.80 

4.28 

14 

17.08 

26 

17.00 

90    6 

34.08 

4.72 

111    4 

19.55 

36 

19.25 

90  16 

38.80 

4.18 

110  54 

21.78 

46 

21.20 

90  26 

42.98 

4.32 

44 

23.92 

56 

23.38 

90  36 

47.30 

Since  the  bubble  readings  which  stand  on  the  same  line 
in  the  second  and  fourth  columns  of  the  table  are  approxi- 
mately equal,  it  is  apparent  that  the  corresponding  circle 
readings  lie  on  opposite  sides  of  A0  and  equally  distant 
from  it.  A0  may,  therefore,  be  determined  by  taking  the 
mean  of  any  pair  of  circle  readings  which  stand  in  the 
same  line,  and  the  angles  A0  —  A',  A"  —  A0,  which  we  shall 
designate  by  T,  may  be  found  by  taking  half  the  difference 
of  corresponding  circle  readings.  Values  of  T  are  given 
in  the  fifth  column  of  the  table. 

The  quantities  2b  are  the  sums  of  the  numbers  in  the 
second  and  fourth  columns,  and  their  differences  given  in 
the  last  column  show  that  any  irregularities  which  may 
exist  in  the  curvature  of  the  level  tube  are  very  small,  and 
we  may  determine  a  mean  value  of  d  to  be  used  over  the 
whole  extent  of  the  level  tube.  Since  the  values  of  r  dif- 
fer so  little  from  90°,  we  may  assume  in  equation  (3) 


and  taking  the  differences  between  the  first  and  fourth,  sec- 
ond and  fifth,  third  and  sixth  lines  of  the  table,  we  shall 
have  A'  —  A"  constantly  equal  to  30',  and  equation  (3)  be- 
comes 


74  BULLETIN    OP    THE    UNIVERSITY    OF    WISCONSIN 


4  tan  y  sin  15'  [1.7959] 


~  2  (6'  -  6")  sin  1*  "      26'  -  26 
from  which  we  obtain  the  following  three  values 

d  =  4. 72  =  4. 74  =  4. 73 

the  mean  of  which  may  be  adopted. 


COMSTOCK — STUDIES    IN   ASTRONOMY  75 


IV.— THE    SIMULTANEOUS    DETERMINATION    OF    FLEXURE, 

INEQUALITY  OF  PIVOTS,  AND  VALUE  OF  A  LEVEL 

DIVISION  FOR  A   "BROKEN"  TRANSIT. 

In  a  "broken"  transit,  i.  e.  one  in  which  the  rays  of 
light  are  bent  at  right  angles  by  a  reflecting  prism  placed 
in  the  axis,  it  is  well  known  that  the  bending  of  the  axis 
under  the  weight  which  it  has  to  carry  produces  an  effect 
upon  the  observed  times  of  transit  of  a  star,  which  may  be 
represented  by  the  expression  f.cos  z  sec  S,  where  /  is  a  con- 
stant peculiar  to  each  instrument,  and  z  and  8  denote  the 
zenith  distance  and  declination  of  the  star.  Since  this  ex- 
pression has  the  same  algebraic  form  as  the  corrections 
for  inclination  of  the  axis,  and  for  inequality  of  pivots, 
they  may  all  be  united  into  a  single  term: 

(V  4-  *+/)  cos  z  sec  d 

where  ±(i+f)  is  a  constant  correction  which  must  be  ap- 
plied to  the  value  of  &'  directly  determined  with  the  spirit 
level.  If  i  +/  is  positive  for  Ocular  West  it  will  be  nega- 
tive for  Ocular  East,  and  the  sign  ±  is,  therefore,  prefixed 
to  it.  Since  it  is  not  necessary  in  the  use  of  a  broken 
transit  to  separate  the  constant  correction  i+f  into  its 
constituent  parts,  it  will  for  the  present  be  treated  as  a 
single  unknown  quantity  whose  value  /?  is  to  be  determined 
in  connection  with  T,  the  angular  value  of  a  half  division 
of  the  level  used  for  measuring  &.  In  a  straight  transit  / 
is  zero,  but  i  has  usually  an  appreciable  value  and  the  cor- 
rection ft  must,  therefore,  be  determined,  and  may  be  con- 
veniently determined  by  the  method  here  developed  for  a 
broken  transit. 

If  from  the  general  equation  of  the  transit  instrument1 
sin  c  4-  sin  8  sin  n  —  cos  d  cos  n  sin  (r  —  m]  =  0  (1) 

i  Chauvenet,  Spherical  and  Practical  Astronomy,  Vol.  II,  §123. 


76  BULLETIN    OF    THE    UNIVERSITY    OF  WISCONSIN 

the  quantities  m  and  n  be  eliminated  by  means  of  the  rela- 
tions (78), l  we  have  the  following: 

sin  c  +  cos  z  sin  b  —  sin  z  cos  b  sin  (a  +  A)  =  0  (2) 

where  90° — a  and  b  represent  the  azimuth  and  altitude  of 
the  point  in  which  the  rotation  axis  of  the  instrument, 
produced  toward  the  west,  intersects  the  celestial  sphere. 
A  and  z  are  the  azimuth  (reckoned  from  the  north  toward 
east)  and  zenith  distance  of  a  star  at  the  instant  of  its 
transit  over  a  thread  whose  collimation  is  c,  i.  e.  the  point 
90" — a,  b  is  the  pole  of  the  small  circle  traced  upon  the 
celestial  sphere  by  the  thread  in  question  when  the  instru- 
ment is  rotated  about  its  axis,  and  the  distance  of  this 
circle  from  its  pole  equals  9(P  +  c. 

Since  in  practice  b  and  c  are  never  so  great  as  10',  equa- 
tion (2)  may  be  written  without  sensible  loss  of  accuracy : 
c  +  cos  z . 6  =  (a  -J-  A)  sin  z  (3) 

Substituting  in  this  equation  for  b  its  value  as  given  by 
the  spirit  level,  and  writing  a  similar  equation  for  the  case 
in  which  the  object  observed  is  not  the  star,  but  its  image 
reflected  from  mercury  or  some  other  level  surface,  we 

have: 

Dir.        c'  +  cos  z'  (n'  T  -f-  /3)  =  (a  +  A')  sin  z'  (4) 

Eef.       c"  —  cos  z"  in"  T  +  /3)  =  (a  +A")  sin  z' 
where  n'  and  n"  are  the  measured  inclinations  of  the  axis 
expressed  in  half  divisions  of  the  level  scale.     We  now  put 

z'  =  z  +  x  z"  =  z  —  x 

and  introducing  these  values  into(4)  find  by  substraction : 

c'  —  c"  +  (n1  +  n')  cos  x  cos  Z.T  -f-  2  cos  x  cos  z  fi 

=  (A'  —A"}  cos  x  sin  z  +  (2a  +  A'  -f-  A")  sin  x  cos  z        (5) 

In  practice  the  object  observed  will  usually  be  a  circum- 
polar  star,    and  owing   to    its    slow   motion   the    quantity 
x  =  J  (z'  —  z")  will  be  so  small  that  we  may  assume 
cos  x  =  1  sin  x  =  cos  S  sin  t  sin  ^  ( T  —  T'} 

where  T'  and  T"  are  the  observed  times  and  t  is  the  hour 
angle  of  the  star  at  the  instant  £  (T'  +  T"). 

*  Loc.  cit. 


COMSTOCK — STUDIES    IN    ASTRONOMY  77 

For  the  coefficient  of  the  last  term  in  equation  (5)  we  ob- 
tain from  (it)  with  sufficient  precision 

2a  -\-  A'  -\-  A*  =  (c  +  c")  cosec  z 
and  introducing  these  values  into  (5)  we  have 

(ri  +  w")  r  -f  2/?  =  (X  —  A")  tan  z  —  (c'  —  c")  sec  z 

-\-  (c'  +  c")  cos  S  sin  t  cosec  z  sin  %  (T  —  T")        (6) 

If  the  star  is  near  the  meridian  or  is  observed  near  the 
collimation  axis  of  the  instrument,  the  last  term  in  this  ex- 
pression will  be  very  small  and  may  frequently  be  neg- 
lected. Putting 

P  =  (A'  —  A")  tan  z 

Q  =  (c  -f  c")  cos  d  sin  t  cosec  z  sin  £  (T  —  T) 

we  obtain  from  the  equations 

sin  z  sin  A  =   —  cos  £  sin  t 

sin  z  cos  A  =.  cos  q>  sin  S  —  sin  q>  cos  $  cos  t  (7) 

reduced  by  means  of  the  relations  furnished  by  the  as- 
tronomical triangle,  the  equation 

P  =  cos  d  cos  q  sec  z  .  2  sin  |  (T  -  T")  206265 

where  q  is  the  parallactic  angle  of  the  star.  Introducing 
Bessel's  auxiliary  JVinto  this  equation,  substituting  in  the 
last  term  of  (6)  in  place  of  cos  3  sin  t  cosec  z  its  equivalent, 
sin  A,  and  collecting  in  a  form  convenient  for  computation 
the  equations  necessary  for  the  reduction  of  a  series  of  ob- 
servations, we  have  the  following: 
tan  N  —  cot  (p  cos  t 

P  =    [-5.615161   cosS     Sin\(T-T"^  (8) 

L  J  sin  z  tan  (N  -{-  d) 

Q  =  (c'  +  c")  sin  A  .sin  $  (T  -  T") 
(n'  4.  n")  T  -f  2/5  =  P  +  Q  —  (c'  —  c")  sec  z 

The  zenith  distance  and  azimuth  of  the  star,  z  and  A  of 
the  formulae,  may  either  be  derived  from  the  instrument 
at  the  time  of  observation,  or  may  be  computed  from  the 
latitude  and  the  co-ordinates  of  the  star,  <?,  s,  t,  by  means 
of  equations  (7). 

Since  ft  changes  sign  when  the  instrument  is  reversed,  a 


78  BULLETIN    OF    THE    UNIVERSITY    OF    WISCONSIN 

similar  pair  of  observations  in  the  reversed  position  will 
furnish  the  equation 

(»'  +  n")  r  -  2fi  =  P  +  Q  -  (c  -  c")  sec  z 

which,  with  the  last  of  equations  (8),  suffices  for  the  de- 
termination of  *  and  ft-  A  large  change  in  the  inclination 
of  the  axis,  e.  g.  one  which  will  give  values  of  n'  and  n" 
with  altered  sign,  may  be  employed  for  the  same 
purpose.  If  the  inclination  of  the  wyes  of  the  instru- 
ment is  not  disturbed  by  the  reversal,  the  level  read- 
ings will  furnish  directly  a  determination  of  the  inequality 
of  pivots,  and  we  shall  have  for  the  flexure 

/  =  ft  -  i 

Formulae  (8)  become  somewhat  simplified  when  the  star 
observed  is  very  near  the  meridian,  but  this  advantage  will 
often  be  outweighed  by  the  convenience  of  observing 
Polaris  at  any  part  of  its  diurnal  path. 

The  application  of  the  formulae  is  illustrated  by  the  fol- 
lowing observations  of  transits  of  Polaris  over  the  microm- 
eter thread  of  a  large  "  broken  "  transit.  Each  observed 
time  and  corresponding  micrometer  reading  is  the  mean  of 
from  five  to  seven  observations  made  in  quick  succession. 
Owing  to  disturbance  of  the  mercury  surface  by  wind,  the 
reflection  observations  were  difficult  and  rather  discordant. 
Since  the  readings  of  the  micrometer  diminish  in  the  di- 
rection of  motion  of  a  star  at  upper  collimation  for  Ocular 
West,  the  collimation  corresponding  to  any  reading,  2i,  of 
the  screw  is  given  by  the  expression 

'  H 

f    -       A-  *V7   W  ( -R        1  ^          +  OcwZar  W- 
t  -    ±  57.57  (R  -  15)         _  Qcular  E 

The  reading  of  the  screw  when  the  threadhis~in  the  collima- 
tion axis  is  assumed  to  be  15.000  rev. 


COMSTOCK — STUDIES    IN    ASTRONOMY 


79 


WASHBURN   OBSERVATORY,   OCTOBER  16,  1894. 
POLARIS   FOR  FLEXURE,    INEQUALITY   OF   PIVOTS,    ETC. 

h.      m.       s. 

a  =     1      20     55.1  q>  =  43°    4'    38* 

d    ==  88°    44'    53*. 5  Chronometer  A  T  ==  +  3.9s. 

Jog  [5.61546]  cos  6  =  3.95484 


OCULAR. 

WEST. 

WEST. 

EAST. 

EAST. 

h.    m.     s 

h.    m.      s. 

h.    m.     s. 

h.    m.     s. 

T 

20    46    20.7 

21      5    52.0 

21    32      8.2 

22      3    54.0 

R' 

11.172 

8.090 

15.274 

15.720 

n'    n" 

+  39.4  +  38.3 

-  42.9  -  42.7 

-  48.9  -  49.4 

+  40.7  +  40.4 

T" 

20    35    57.4 

21     14    16.7 

21    40    39.3 

21    53    50.7 

R" 

12.800 

6.289 

17.431 

13.060 

t 

19    20    17.8 

19    49    13.2 

20    15    32.6 

20    38      1.1 

z 

46    30    15 

46    21    35 

46    14      0 

46      8      5 

A 

1     37 

1     32 

1    26 

1    20 

log  cos  t 

9.53558 

9.66155 

9.74628 

9.80355 

N 

20      9    25 

26      7    56 

30    48    25 

34    13    43 

log  cosec  z 

0.13941 

0.14045 

0.14136 

0.14208 

logcot(N+d) 

9.53463n 

9.66630n 

9.75362n 

9.81213n 

log  sin  |  (T  —  T"} 

8.35530 

8.26364n 

8.26911n 

8.34113 

log  sin  A 

8.452 

8.429 

8.400 

8.369 

log  (c'  +  c") 

2.541n 

2.955n 

2.193n 

1.848 

logP 

1.98618n 

2.02523 

2.11893 

2.25018n 

logQ 

9.348n 

9.648 

8.862 

8.558 

log  (c1  —  c") 

1.97185n 

2.01571 

2.09405 

2.18508n 

log  sec  z 

0.16222 

0.16107 

0.16007 

0.15929 

(c'  —  c")  sec  z 

-  136.17 

+  150.24 

+  179.52 

-  220.99 

P 

-    96.87 

+  105.98 

+  131.50 

-  177.90 

Q 

-      0.22 

+     0.44 

4-      0.07 

+     0.04 

80  BULLETIN    OF    THE    UNIVERSITY   OP    WISCONSIN 

The  preceding  computation  furnishes  the  absolute  terms 
of  the  following  equations: 

• 
+  77.7r  +  2^  =  +39.08 

-  85.6  r  + 2/5  =   -43.82 

-  98.3  r  —  2/3  =   -  47.95 
+  81.1  r  -  2fi  =  +  43.13 

A  least  square  solution  of  these  equations  furnishes  the 
Talues : 

if  jr 

r  =  +  0.506  ft  =   -  0.600 

Prom  numerous  determinations  with  the  spirit  level,  the 
inequality  of  the  pivots  is  known  to  be  i  —  —  0".64,  which, 
combined  with  the  value  of  A  gives  for  the  flexure  the 
Talue/=  +  (T.04. 


COMSTOCK — STUDIES    IN    ASTRONOMY  81 


V.  — DETERMINATION   OF   TIME  AND  AZIMUTH   FROM  TRAN- 
SITS OVER  THE  VERTICAL,  OF  THE  POLE  STAR. 

In  a  development  of  the  formulas  for  determining  the 
time  from  transits  over  the  vertical  of  a  circum-polar  star, 
published  in  1828,  Bessel  says  by  way  of  introduction: 
"  That  this  may  not  appear  futile  I  remark,  what  Hansteen 
and  Schumacher  have  properly  noted,  that  the  most  ap- 
propriate use  of  a  portable  transit  instrument  for  a  time 
determination  consists  in  mounting  it,  not  in  the  meridian, 
but  in  an  azimuth  which  admits  of  an  observation  of  one  of 
the  polar  stars,  wherever  this  may  be  with  respect  to  the 
meridian,  closely  followed  or  preceded  by  a  transit  of  a 
fundamental  star." 

The  obvious  advantage  which  this  mode  of  observing 
possesses  lies  in  the  shorter  period  of  time  during  which 
the  observer  depends  upon  the  stability  of  his  instru- 
mental constants.  For  meridian  observations  this  period 
is  rarely  much  less  than  half  an  hour,  while  by  the  method 
suggested  it  need  never  exceed  five  minutes.  Nevertheless, 
the  general  opinion  of  two  generations  of  field  astronomers 
seems  fairly  represented  by  the  words  of  Chauvenet,  who, 
after  devoting  a  score  of  pages  to  a  discussion  of  the 
method,  remarks  in  closing:  "The  methods  which  have 
here  been  given  *  *  *  are  intended  for  the  use  of  ob- 
servers in  the  field  who  have  but  little  time  to  adjust  their 
instruments  and  wish  to  collect  all  the  data  possible,  re- 
serving their  reduction  for  a  future  time.  The  greater 
labor  of  these  reductions,  compared  with  those  of  meridian 
observations,  is  often  more  than  compensated  by  the  saving 
of  time  in  the  field. "  This  greater  labor  of  reduction  is 
now  obviated  through  the  simplifications  introduced  into 
the  method  by  the  Russian  astronomer,  Dollen,  who  main- 
tain s  with  equal  zeal  and  cogency  the  greater  precision  and 


82  BULLETIN    OF    THE    UNIVERSITY   OF    WISCONSIN 

at  least  equal  convenience  of  his  method  for  all  purposes 
of  field  astronomy.  Under  Dollen's  influence  the  method 
has,  within  the  last  quarter  century,  come  into  consider- 
able use  in  eastern  and  central  Europe,  and  from  an  ex- 
tended practical  application  of  it  the  writer  of  these  pages 
is  satisfied  of  the  justice  of  the  claims  made  in  its  behalf. 
This  section  of  the  present  paper  is  an  attempt  to  bring  to 
the  attention  of  American  teachers  of  practical  astronomy, 
in  substance,  the  theory  of  Dollen's  method,  but  it  cannot 
be  considered  a  substitute  for  the  precepts  and  discussion 
contained  in  the  elaborate  introduction  to  the  Stern  Ephem- 
eriden  zur  Bestimmung  von  Zeit  und  Azimut,  published  annu- 
ally by  Dollen  since  1886. 

As  indicated  by  the  above  title,  the  observations  for 
time  are  equally  available  for  a  determination  of  azimuth, 
and  reduced  to  their  simplest  terms  these  observations  are 
as  follows:  Let  the  transit  (universal  instrument,  or  the- 
odolite, in  case  a  determination  of  azimuth  is  also  desired) 
be  pointed  at  Polaris,  and  the  chronometer  time,  S',  at 
which  the  star  appears  bisected  by  the  middle  vertical 
thread,  noted.  Then  revolve  the  telescope  about  the  hori- 
zontal axis  without  disturbing  the  azimuth  of  the  instru- 
ment and  observe  the  time  of  transit,  S,  of  a  clock  star  over 
all  of  the  threads,  and  measure  the  inclination  of  the  axis, 
&,  with  a  spirit  level,  if  possible  both  before  the  observa- 
tion of  Polaris  and  after  that  of  the  southern  star.  Reverse 
the  instrument,  point  again  upon  Polaris,  and  observe  it 
and  a  clock  star,  as  before.  If  the  instrument  possess  a 
graduated  horizontal  circle,  which  is  read  in  connection 
with  the  observations  of  the  stars,  these  data  will  deter- 
mine the  zero  point  of  the  circle,  i.  e.  its  reading  when  the 
telescope  points  north,  and  the  azimuth  of  any  terrestrial 
point  toward  which  the  telescope  may  be  directed. 

We  proceed  to  consider  the  theory  of  the  method  and 
adopt  as  a  basis  for  the  investigation  the  fundamental 
equation  of  the  transit  instrument,1 

i  Chauvenet,  Vol.  II,  Eq.  (79). 


COMSTOCK — STUDIES    IN    ASTRONOMY  83 


sin 


\in  (r  —  m)  =  tan  n  tan  d  -J-  sin  c  sec  n  sec  8  (1) 

together  with  the  equations 

tan  n  =  sin  b  sec  n  cosec  q>  —  sin  m  cot  <p         (2} 
cos  a  tan  m  =  tan  b  cos  g>  -f-  sin  <p  sin  a          (3) 

furnished  by  the  spherical  triangle.  PZA,  formed  by  the 
pole,  the  zenith  and  the  point  in  which  the  rotation  axis 
of  the  instrument,  produced  toward  the  west,  intersects 
the  celestial  sphere.  The  sides  and  angles  of  this  triangle 
have  the  following  values: 

PZ  =  90°  -  q>    PA  =  90°  -  n    ZA  =  90°  -  b 
P  ==  90°  -  m    Z  =  90°  +  a 

The  symbol  r  represents  the  east  hour  angle  of  the  star 
at  the  instant  of  transit  over  the  middle  thread,  and  we 
have  obviously  the  relation 

r=a-S-JT  (4) 

Since  each  star  observed  furnishes  an  equation  of  the 
types  (1)  and  (4\  it  appears  that  if  the  instrumental  con- 
stants 1}  and  c  are  known  an  observation  of  the  transits  of 
a  circum-polar  star  and  a  southern  star  suffice  for  the  de- 
termination of  the  unknown  quantities  AT,  m,  n,  a,  and  our 
problem  consists  solely  in  so  transforming  the  preceding 
equations  as  to  facilitate  the  determination  of  AT  and  a. 

Denoting  by  the  subscripts  1  and  2,  respectively,  quanti- 
ties pertaining  to  the  polar  and  the  southern  star,  we 
write  equation  (1)  for  each  of  these  stars  as  follows: 

sin  (z-j  —  m  —  3)  =  tan  dl  tan  n  \  1  -f  cosec  S:  cosec  n  sin  (c  -\-  x^)  I 

(5) 
c  i 

sin  (r2  —  m  —  5)  =  tan  S2  tan  n  •]  1  -f-  cosec  d2  cosec  n  sin  (c  -f-  #3 


where  3,  xlt  and  x2  are  small  arbitrary  quantities  subject 
only  to  the  condition  that  they  must  be  so  determined  as 
to  satisfy  the  equations.  Since  this  is  equivalent  to  only 
two  relations  among  the  three  quantities  we  are  at  liberty 
to  impose  a  third  relation,  for  which  we  choose 
sin  (c  -f-  #1)  sin  S2  =  sin  (c  +  #3)  sin  £t 


84  BULLETIN    OF    THE    UNIVERSITY   OF    WISCONSIN 

which  makes  the  bracketed  factors  in  the  two  equations 
equal.  Presupposing  that  $,  xly  and  x2  are  small  quantities 
we  differentiate  equations  (5),  and  eliminating  xlt  and  x2 
find,  when  quantities  of  the  order  en2  are  neglected, 

^  =  __  (1  -  sin  d2)  c  _ 
cosd2  —  sin  62  cotS1  cos  (rl  —  m) 

If  for  $2  we  substitute  the  polar  distance,  P3  =  90°-S2, 
this  equation  becomes,  very  approximately, 

3-  =  c  .  tan  \pz    \  I  +  cot  d1  tan  S2  cos  (T±  —  m)  I  (6) 

Dividing   the  first  of  equations   (o)   by   the   second,  we 
obtain  : 

'       tan     *  (,,  +  r.)  -•-»-  +        *»  i  (r,  -  r.)       (7) 


We  now  assume  the  auxiliary  quantities, 

2r  =  (a±  -  S')  -  (aa  -  S) 
U  '  =  a2  -  S  -  AT  -  m  -  5  (8) 

and  introducing  them  into  (7)  find 

,     ,    ,.,,         sin  ($..  +  £2)  . 
ton(r+t7)=-i||(a'i  +  a->to»r 

whose  solution  is 

tan  U  =       cot  6  ^  tanS,  tin  2r 

1  —  cot  dl  tan  62  cos  2r 

In  equations  (£)  dT-\-  m  is  now  the  only  unknown  quantity, 
and  to  determine  m  we  apply  (1)  to  the  polar  star  and  sub- 
stitute in  it  the  value  of  tann  given  by  (2)  and  the  value 
of  T^  —  m  given  by  (4)  and  (8),  and  find 

sin  m  =  —  cot  di  tan  g>  sin  (2r  -f-  U  +  $)  4~  sin  b  sec  (p  +  sin  c  tan  <p 
in  which  terms   of  the  order  en*  are  neglected.     Subtract- 
ing from  each  member  of  the  equation  the  auxiliary  quantity 
sin  m'  =   —  cot  d^  tan  cp  sin  (2r  -f-  U)  (10), 

we  obtain  to  the  same  .degree  of  approximation 

m  =  m'  -\-  b  sec  (p  -f-  c  tan  cp  —  5  cot  d^  tan  <p  cos  (2r  +  U) 

Substituting  for  3-  its  value  in  terms  of  c,  and  introduc- 
ing into  (8)  the  resulting  value  of  m,  we  obtain 

4T+Cc  =  aa-  (S+U+m'  +  b  sec  <p}  (11) 


COMSTOCK — STUDIES    IN    ASTRONOMY  85 

where  the  coefficient  C  has  the  value 

C  =  tan  <p  +  tan  |pa  ]  1  +  (tan  d2  -  tan  g>)  cot  <J±  cos  (2r  -f  U)  I    (12) 

If  at  the  time  of  observation  the  southern  star  was  near 
the  zenith,  or  Polaris  was  near  elongation,  or  the  collima- 
tion  constant,  c,  was  very  small,  the  bracketed  factor  may 
be  put  equal  to  1,  giving 

C  =  tan  <p  +  tan  %  p2 

For  a  determination  of  azimuth  we  write  equation  (3)  in 
the  form 

tan  a  =  tan  m  cosec  g>  —  tan  b  cot  g> 

and  assuming  the  equation 

tan  a'  =  tan  m'  cosec  q>  (13) 

find  by  subtraction 
a  =  a'  +  b  tan  <p  -\-  c  sec  g>  \  1  —  cot  d^  tan  %  p2  cos  (2r  +  C7)  j-     (14) 

If  K  and  M  denote  respectively  the  reading  of  the  azi- 
muth circle  corresponding  to  the  star  observations,  and  to 
that  position  of  the  instrument  in  which  the  rotation  axis 
lies  in  the  plane  of  the  prime  vertical  (collimation  axis  in 
the  meridian),  we  have,  obviously, 

M  =  K  +  a'  +  b  tan  cp  +  C'c  (15) 

where  C'  is  an  abbreviation  for  the  coefficient  of  c  given  in 
the  preceding  equation. 

Since  the  collimation  constant,  c,  changes  sign  when  the 
instrument  is  reversed,  an  observation  of  Polaris  and  a 
southern  star  in  each  position  of  the  instrument,  W.  and 
E.,  will  suffice  for  the  determination  of  4  T  and  c  from  the 
observed  times  of  transit,  and  also,  if  the  instrument  is 
provided  with  an  azimuth  circle,  for  the  determination  of 
M  and  c,  from  the  circle  readings.  The  agreement  between 
the  two  values  of  c  thus  determined  furnishes  a  valuable 
control  upon  the  accuracy  of  the  observations  and  their 
reduction. 

In  the  preceding  investigation  the  effect  of  flexure,  ine- 


86  BULLETIN    OF    THE    UNIVERSITY   OF    WISCONSIN 

quality  of  pivots  and  diurnal  aberration  has  been  neg- 
lected. These  quantities  may,  however,  be  taken  into  ac- 
count, as  in  the  case  of  meridian  observations,  by  applying 
to  the  observed  level  constant,  &,  a  correction,  ±  p,  for 
the  first  two  sources  of  error,  and  by  applying  to  S  a  cor- 
rection, 

s. 
-  0.021  cos  (p.  C 

for  the  aberration. 

The  formulae  requisite  for  the  reduction  of  observations 
in  the  vertical  of  the  pole  star  may  now  be  collected, 
slightly  simplified  and  arranged  as  follows : 

Data  known  independently  of  the  observations: 

8. 

<p,  a19  a2,  8j,  82,   K  —  0.021  cos  q>,   pa   =  90°  -  S2 
Data  given  by  the.  observations:     S",  S,  6,  K. 

t  =  (a i  -  or,)  +  (S  -  S') 
h  =  I  -f-  tan  Ss  cot  dl  cos  t 
1  =  1  —  tan  \pz  cot  8l  cos  t 
C  =  h  tan  £  p2  +  /  tan  cp 
C'   =   15  I  sec  <p 


tan  U  = 


cot  8±  tan  ds  sin  t 


1  —  cot  dt  tan  62  cos  t 
—  sin  m'  =  tan  g>  cot  8t  sin  (t  -f  U} 

tan  a'  =  tan  m'  cosec  (p 

A  T  +  Cc  =  <xa  -  (S  +  U  +  m'  -f  b  sec  q>  -  CK) 
M  -  C  c  =  K  -f  a'  +  b  tan  <p  -  C'x 

The  computation  of  these  formulae  may  be  somewhat  fa- 
cilitated by  an  algebraic  device  upon  which  Dollen  places 
great  stress.  From  the  ordinary  development  of  sin  x  and 
tan  x  in  series,  we  have,  when  x  is  small, 

log  sin  x  =  log  x ~-  log  tan  x  —  log  x  -f  2  -~- 

where  M  denotes  the  modulus   of  the  common  system  of 
logarithms.     Putting 

6  =  $  MOJ« 

we  may  tabulate  6  with  x  or  log  x  as  argument,  and  such  a 


COMSTOCK — STUDIES    IN    ASTRONOMY  87 

table  is  given  by  Dollen  with  log  x,  when  x  is  expressed  in 
seconds  of  time,  as  argument.  When  x  is  expressed  in  arc 
values  of  6  may  be  taken  from  any  logarithmic  table  by 
means  of  the  relation 

6  =  i  (log  tan  x  —  log  sin  x) 

If  d(U)  denote  the  value  of  6  corresponding  to  log  U 
when  Uis  expressed  in  seconds  of  time  we  may,  by  the 
introduction  of  the  divisor,  15  sin  Y ,  obtain  in  seconds  of 
time  and  arc,  respectively, 

U  =    [4.13833  -  26  (U)l       cot  S    tan  S    sin  t 
L  J   1  —  cot  6\  tan  d3  cos  t 

r  n  <17) 

(-  m'}  =    I  4.13833  -f  6  (ra')J  cot  d±  tan  q>  sin  (t  -f-  U") 

log  a'  ==  log  (15  cosec  q> )  +  log  m'  -\- 2  6  (m')  —  26  (a') 

In  equations  (16)  the  quantities  h,  I,  C,  C'  are  analogous 
to  the  transit  factors  A,  B,  C  used  for  the  reduction  of  me- 
ridian observations,  and  (7,0"  may  be  tabulated  for  a  given 
latitude  and  assumed  constant  for  a  period  of  several  years. 
The  quantities  U  and  m'  must  be  computed  anew  for  each 
observation,  and  a'  must  also  be  computed  in  case  the  azi- 
muth is  required.  To  diminish  the  labor  of  this  computa- 
tion Dollen  tabulates  for  a  selected  list  of  180  stars  certain 
General  Constants,  through  which  these  computations  are 
considerably  shortened. 

With  assumed  values  of  the  coordinates  of  the  stars  and 
an  assumed  interval  S — 8'  —  4m  put 

TT  i    206*265         »  __ 

—  U  =   x0  -\ — —  cot  d±  sin  (t-}-  U)   =  NQ 

10 

We  shall  then  have 

-  (U+  m'}  =  x0  +  p  N0   =  t0  a'  =  p'  N0 

where  p  and  p'  are  functions  of  the  latitude  which  differ 
from  tan  <p  and  15  sec  9  by  terms  of  the  kind  above  repre- 
sented by  6, 

log  p  =  log  tan  g>  +  6  ( N  tan  <p) 
logp'  =  log  (15  sec  cp)  +  3  6  (N  tan  q>)  —  2  6  (.Vsec  (p) 


BULLETIN    OF    THE    UNIVERSITY   OF    WISCONSIN 


The  values  of  p  and  p'  may  be  conveniently  tabulated  for 
a  given  latitude  with  log  N  as  the  argument,  and  for  this 
purpose  log  p'  is  best  expressed  in  the  form 

log  p'  =  log  (15  sec  fp]  +  d  (N  V  tan*q>  —  2) 

where  the  two  6  terms  given  above  have  been  united  into 
a  single  term  whose  numerical  value  is  to  be  obtained  re- 
gardless of  the  sign  of  the  quantity  under  the  radical,  and 
then  to  be  added  or  subtracted  as  this  quantity  is  positive 
or  negative.  The  following  is  such  a  table  for  the  latitude 
of  the  "Washburn  Observatory,  <p  =  43°  4'  37",  and  it  should 
also  be  noted  that  the  values  of  log  N  are  limiting  values  at 
which  the  tabular  p,  p'  changes  from  one  value  to  the  next: 


P 

logN 

.97083 

2.238 

.97084 

2.381 

.97085 

2.476 

.97086 

2.539 

.97087 

2.587 

.97088 

P' 

logN 

1.31251 

1.921 

.31250 

2.239 

.31249 

2.363 

.31248 

2.442 

.31247 

2.499 

.31246 

The  construction  of  such  a  table  is  the  only  point  at 
which  the  6  terms  are  required  in  the  application  of  Dol- 
len's  ephemerides. 

In  general  the  coordinates  of  the  stars  and  the  observed 
interval  8 — 8'  will  differ  from  that  assumed  in  the  compu- 
tation of  ojo'and  JVj  and  it  will  be  convenient  to  pass  from 
these  latter  quantities  to  the  values  a?,  ^corresponding  to 
the  actual  observation  by  means  of  differential  formulae. 

It  is  evident  from  an  inspection  of  equations  (16)  that 
these  differential  formulas  will  contain  some  terms  which 
involve  only  the  coordinates  of  the  stars  and  are,  therefore, 
the  same  for  all  parts  of  the  earth's  surface,  while  other 
terms  will  involve  functions  of  the  latitude,  and  only  that 
part  of  these  terms  which  is  independent  of  the  latitude 
can  conveniently  be  tabulated.  Leaving  the  reader  to  dif- 


COMSTOCK  —  STUDIES    IN    ASTRONOMY  89 

ferentiate  for  himself  equations  (16),  we  reproduce  here 
the  form  in  which  Dollen  expresses  the  differential  coeffi- 
cients and  the  correction  terms  involving  them: 

x+pN  =  f0  +  Qk  +  RG  +  DA  d 
a'  =  p'N  =--  p'NQ  +  Q'k  +  R'G 

where  Jc,  G  and  A  S  represent  variations  in  the  elements 
with  which  x0  and  NQ  were  computed,  and  Q,  Q',  R,  R',  D 
are  differential  coefficients  having  the  following  values: 

Q  =  pX  -f  ft  R  =  pjii  -f  y 

q  =  p'\  R'  =  p> 

The  values  of  0,r  •*-,/*  andD  involve  only  the  coordinates 
of  the  stars  and  are  given  among  the  general  constants  for 
each  star  of  Dollen's  list. 

The  values  of  fc,  G  and  Ad  are  as  follows: 


A  a  =  a2  -  (or2)0  g  =   -     a^  -  (ajo   - 

G  - 
Ad    =   S2  -  (5,)0  fc    =    -     S    - 


where  the  subscript  0  denotes  the  tabular  values  of  the  co- 
ordinates corresponding  to  x0,  N^.  These  assumed  values 
are  given  as  a  part  of  the  table  of  constants  for  each  star, 
and  an  ephemeris  of  g  and  log  k  precedes  the  table  of  con- 
stants. 

The  actual  reduction  of  a  set  of  observations  by  means  of 
these  general  constants  will  not  often  be  made,  but  recourse 
will  be  had  to  the  General  Ephemerides  constructed  from 
them  for  93  of  the  180  stars.  These  ephemerides  give  at 
intervals  of  ten  days  throughout  the  year  the  instantaneous 
values  of  ^V  and  T,  T  =  a%  -f-  x,  and  from  them  the  observer 
should  construct  a  local  epherneris  of  the  values  of  0  and 
a'  for  a  few  of  the  tabular  dates  near  the  epoch  of  his  ob- 
servations, using  the  relations 

9  =   T  +  pN          a'  =  p'N 

Values  of  0  and  a'  interpolated  from  the  ]ocal  ephemeris 
will  be  immediately  available  for  the  reduction  of  observa- 
tions in  which  the  observed  interval  S  —  S'  equals  the 


90  BULLETIN    OP    THE    UNIVERSITY    OF    WISCONSIN 

value  4m  assumed  in  the  computation  of  XQ  and  NQ  The  ob- 
servations should  be  so  arranged  as  to  secure  at  least  a 
rough  approximation  to  this  interval  between  the  observa- 
tion of  Polaris  and  the  clock  star,  but  a  deviation  of  even 

f 

several  minutes  from  the  prescribed  amount  may  be  very 
simply  corrected. 

Since  the  interval  S —  S'  affects  U,  m'  and  a'  precisely  as 
does  ^  —  a2  whose  effect  is  represented  in  the  term  EG, 
we  apply  to  S  and  K  the  corrections 


R  J  S  -  (S'  +  4»»)  £  ^'  j  S  -  (S'  -f  4m) 

and  the  reduction  of  the  observations  takes  the  very  sim- 
ple form : 


r  — 

80   =  S  +  R0r  +  £6  -  CH  AT  ±  Cc  =  6  -  S0 

K0   =  K  -f-  R'Qr  +  B'b  -  C'x  M  T  C'c  =  #0  -  a' 

The  level  corrections  Bb,  B'b  are  most  conveniently  taken 
from  a  table  of  multiples  of 

— -  sec  <p  =  Br  -  tan  CD  --  B'r 

do  2 

where  r  represents  the  angular  value  of  one  division  of 
the  level  scale.  The  factor  EQ  equals  100  R  and  its  value 
together  with  that  of  the  collimation  factors  (7,  C'  are  to  be 
derived  from  the  data  given  with  each  star  in  the  ephemeris 

I?'  l^\'   I  I  /"V  /v-k'/^V 

JK     Q         P  //  Q  U  =       P    O    I 

These  values  when  once  computed  should  be  preserved  for 
future  use. 

The  reduction  to  the  middle  thread  of  transits  of  a  clock 
star  observed  over  the  side  threads  must  not  be  made,  as 
in  the  meridian,  by  the  use  of  the  factor  (7,  but  by  a  special 
factor  F  whose  logarithm  is  given  in  the  ephemeris  and 
among  the  general  constants  for  each  star. 
F  =  sec  d»  sec  n  sec  r 


COMSTOCK — STUDIES    IN    ASTRONOMY  91 

Certain  auxiliary  quantities  to  be  used  in  setting  the  in- 
strument so  as  to  find  the  stars  to  be  observed  are  also 
given  in  the  tables.  Their  use  will  be  understood  from 
the  following  precept:  "At  the  sidereal  time  e — 4m  point 
upon  the  pole  star  by  means  of  its  azimuth  a'  and  zenith 
distance  z'  =  H—  (<?+  v  tan  <p)  and  without  changing  the 
azimuth  of  the  instrument  await  the  clock  star  at  the 
zenith  distance  z  —  p  —  z- " 

The  following  two  examples  illustrate,  respectively,  the 
application  of  the  trigonometric  formulas,  equations  (16} 
and  (17),  and  of  Dollen's  ephemerides,  to  the  reduction  of 
observations  made  with  a  very  small  universal  instrument, 
having  an  objective  with  a  clear  aperture  of  35  mm,  focal 
length  373mm,  magnifying  power  of  ocular  36  diameters,  azi- 
muth circle  read  by  estimation  to  single  seconds.  In  view 
of  the  small  dimensions  and  feeble  power  of  the  instru- 
ment the  agreement  between  the  values  of  the  collimation 
constant  c  given  by  the  observed  times  and  the  circle  read- 
ings is  sufficiently  satisfactory. 

The  computation  by  the  trigonometric  formulse  is  so  ar- 
ranged that  the  values  of  U,  m',  etc.,  may  be  obtained 
either  with  or  without  the  use  of  the  6  terms. 


92  BULLETIN    OF    THE    UNIVERSITY   OF    WISCONSIN 

1891,  SEPTEMBER  4.      OBSERVER,  G.  C.  C. 

BAMBERG  UNIVERSAL  INSTRUMENT. 


h.  m.    s.                             ° 

<p  =  43    4    47             <*!   =  1     19    38              8±   =  88    43    84.7 

log  tan  q>  —  9.97082         log  cosec  <p  =  0.16559         log  cot  d^   =  8.34702 

Star.  Oc. 

e  Cygni  W. 

ZCygniE. 

«i  ~  <** 

4  37  43     4  11  13 

az 

20  41  50.19 

21    8  20.00 

S  -  S' 

4  28           4  35 

8, 
C     C' 

+  33  33  57 
1.47    20.7 

+  29  47  1 
1.51    20.7 

t 
cos  t 

70  32  45 
9.52251 

63  57    0 
9.64262 

S' 

20  38  56 

21    5  39 

tan  32 

9.82187 

9.75764 

s 

20  43  23.59 

21  10  14.03 

sin  t 

9.97447 

9.95348 

b  .  I)  sec  <p 

-  7.2  -  0.65 

s. 
-f  5.1  -f  0.46 

(cot  8j_  tan  S2  cos  t) 

7.69140 

7.74738 

U 

-f  3  12.22 

+  2  38.08 

l-(    ) 

9.99786  i  9.99757 

m' 

-  4  30.87 

0                                       " 

-  4  18.27 

Q                                           » 

cot  #!  tan  8  2  sin  t 

8.14336 

8.05814 

CK    C'x 

0. 

0.02    0.3 

o* 

0.02    0.3 

tan  U 

8.14550 

8.06057 

K 

344  38  31.5 

154  33  44.0 

26 

3 

2 

a' 

-  1  39  8.1 

-  1  34  31.6 

sin  (t  +  U) 

9.97656 

9.95588 

b  tan  <p 

-  6.7 

+  4.8 

—  f>in  m' 

8.29440 

8.27372 

AT  ±  Co 

-  14.08 

-  14.28 

6 

3 

3 

M  T  C'c 

332  59  16.4 

152  59  16.9 

tan  a' 

8.46008 

8.43939 

s.                 " 
AT  +  1.47  c  =   -  14  08            M  -  20  7  c  =  16.4 

AT  -  1.51  c  =   -  14.28            If  +  20.7  c  =  169 

s. 
AT  =    -  14.18                          J/  =  16.6 

c=   +    0.07 


c  =        0,01 


COMSTOCK — STUDIES    IN    ASTRONOMY 


93 


1891,  SEPTEMBER  4. 

logp  =  9.97085  log  p'  =  1.31247 


Star  Oc. 

s  Cygni  W. 

C  Cygni  E. 

Equations: 

R    R 

-0.196  -14.85 

-0.350  -19.85 

s. 

C    C' 

1.471    20.46 

1.513    20.42 

JT+1.47c   =  -  14.08 

B 

20    43    8.79 

21    10    0.06 

T-  1.51  c    =   -  14.29 

b 

-  7.2 

+  5.1 

AT  =  -  14.18 

S'  +  4™ 

20    42    56 

21      9    39 

c  =  +   0.07 

8 

20    43    23.59 

21    10    14.03 

M"  -  20.  5  c  =  16.3 

CK.  RQr 

-  0.02    -  0.05 

-  0.02    -  0.12 

Jlf"  +  20.4c  =  16.7 

b  sec  <p 

-  0.65 

+  0.46 

M    =  16'.  5 

AT  ±  Cc 

-  14.08 

-  14.29 

c  =  +0.01 

a1 

O           '             » 

1    39    4.1 

0           '              it 

I    34    24.8 

O           '              it 

H  =  332    57    16.5 

K 

334    38    31.5 

154    33    44.0 

C'n    R'r 

-  0.3    -  4.1 

-  0.3    -  7.0 

b  tan  (p 

-  6.7 

+  4.8 

M=f  C'c 

332    59    16.3 

152    59    16.7 

94  BULLETIN   OF    THE    UNIVERSITY   OF    WISCONSIN 


VI.— DETERMINATION   OF  LATITUDE  AND   TIME  FROM 
EQUAL  ALTITUDES  OF  STARS. 

The  simultaneous  determination  of  time  and  latitude  from 
the  observed  instants  at  which  three  different  stars  reach  the 
same  (unknown)  altitude  is  discussed  in  the  principal  text 
books  of  spherical  astronomy,  but  the  laborious  character 
of  the  reduction  of  the  observations  there  developed  has 
prevented  the  method  from  coming  into  general  use,  al- 
though from  theoretical  considerations  and  from  experi- 
ence it  has  been  abundantly  shown  to  furnish  a  very 
accurate  determination  of  both  time  and  latitude.  In  the 
following  pages  an  attempt  is  made  to  simplify  the  method 
by  substituting  for  the  observation  of  three  stars  separated 
by  considerable  intervals  of  time  the  observation  of  the 
time  at  which  a  single  star  transits  over  the  almucantar  of 
a  close  circum-polar  star,  usually  Polaris,  the  elapsed  time 
between  the  pointing  of  the  instrument  upon  the  polar 
star  and  the  observed  transits  of  the  clock  star  being 
made  as  short  as  possible,  e.  g.  five  minutes,  or  less. 

Such  a  comparison  of  one  clock  star  with  one  polar  fur- 
nishes a  single  relation  between  the  latitude  and  the  clock 
correction,  and  a  similar  comparison  of  another  star  fur- 
nishes a  second  relation  which  suffices  for  the  determina- 
tion of  both  quantities.  It  should  be  noted  that  these  two 
sets  of  observations  are  entirely  independent  of  each  other 
and  require  no  assumption  with  regard  to  the  stability  of 
the  instrumental  constants,  save  for  the  brief  interval  be- 
tween pointing  upon  Polaris  and  observing  the  southern 
star. 

The  almucantar  and  the  zenith  telescope  are  the  instru- 
ments best  adapted  to  observations  of  this  kind,  but  any 
instrument  which  possesses  a  telescope  rotating  about  a 
horizontal  and  a  vertical  axis  and  provided  with  a  level 
whose  plane  is  perpendicular  to  the  horizontal  axis,  may  be 


COMSTOCK — STUDIES    IN   ASTRONOMY  95 

used,  e.  g.  a  universal  instrument  or  an  engineer's  transit. 
If  the  makers  would  furnish  a  simple  means  of  fastening 
the  striding  level  which  accompanies  the  better  class  of 
transits,  with  its  tube  at  right  angles  to  the  horizontal 
axis,  the  efficiency  of  these  instruments  would  be  very 
greatly  increased,  but  even  without  this  attachment  the 
observation  of  equal  altitudes  is  the  most  advantageous 
mode  of  employing  such  an  instrument  for  the  determina- 
tion of  either  latitude  or  time.  We  proceed  to  develop  the 
equations  for  the  general  case  in  which  both  of  the  quan- 
tities are  required. 

Let  Tl  and  T2  denote  the  observed  times  at  which  two 
stars  cross  a  given  almucantar  whose  (unknown)  zenith 
distance  is  2,  and  let  a^,  it,  aa,p,  be  the  right  ascensions  and 
polar  distances  of  the  northern  and  southern  star,  respec- 
tively. The  formulae  for  the  transformation  of  coordinates 
furnish  for  the  two  stars  the  equations : 

cos  z  =B  sin  (f>  cos  it  -|-  cos  g>  sin  it  cos  (T  +  r) 

(1) 
=  sin  (p  cos p  +  cos  <p  sin p  cos  (T  —  T) 

where 

T+r  =   T±  +  AT  -  «t  T  -  r  =   T2  +  AT  -  az 

Subtracting  the  second  equation  from  the  first  and  divid- 
ing by 

2  sin  %  (p  +  TT)  sin  £  (p  —  -rt)  cos  (p 
we  obtain 

tan  (f>  ==  cot  $  (p  -{-  K)  cos  T  cos  r  —  cot  ±  (p  —  TT)  sin  T  sin  r        (2) 

We  introduce  into  this  equation  the  auxiliaries 

I  cos  A  =  cot  ^  (p  -}-  7t)  cos  r     I  sin  A  =  cot  ^  (p  —  ft)  sin  r   (3) 

and  obtain 

lcos(T-  A)  =  tan  <p  (4) 

From  equations  (3)  we  obtain 

I  sin  (A  —  r)  ==  J  cof  £  (p  —  ft]  —  cot  ^  (p  +  "0  [  sin  r  cos  r 
I  cos  (A  —  r)  =   cotf  |  (p  —  it]  sin*  T  +  cot  $  (P  —  ^  cos2  r 


96  BULLETIN   OF    THE    UNIVERSITY   OP   WISCONSIN 

which  furnish,  after  a  little  reduction, 

tan  (A  -*  4  -     .     Sin7tsin^ 

sin  p  —  sin  7t  cos  2r 

We  also  obtain  from  (3) 


,  L  _.  _       _  ~ 

cos  r  ~  sin  T  )       ~  si/i  i  (p  +  TT)  si/i  £  (p  —  n) 


from  which 


,  ! cos  7t  —  cos  p  ^  sin  (r  -f-  A) 

sin  p  sin  2t 


In  this  expression  we  put 

COS  7t  —  COS  p 

-  :  -  —  —  tan  i  (  p  —  x) 
sin  p 

and  find  the  rigorous  equation 

tan  $  x  =  fan2  \ncot\p  (9) 

for  which  there  may  usually  be  substituted 

2  sin2  i  it 

X    ~-    -  :  —  -4  —  COt  i  p 

sin  1 
Introducing  (8)  into  (4)  it  becomes 

cos  (T  -  A)  =  tan  9  tan  \(p-x)  S-^~-  (10) 

We  now  put 

X-  T  =  M  !T-A  =  iY 

and  obtain 

T  -  r  =   T2  +  4T-  aa  =  Jf-f  N  (11) 

These  equations  suffice  for  the  determination  of  ^T  when 
the  latitude,  <p,  is  known,  and  the  effect  upon  ^T  of  an  er- 
ror in  the  assumed  value  of  <p  is  readily  shown  to  be 

^A-  4  q>  =  y£.  z/  (p  =  -  2  cosec  2  ?>  cof  N.  A  <?  (12) 


Putting  t  —  2r  and  eliminating  the  formulae  requisite  for 
the  reduction  of  an  observation  may  be  collected  and  ar- 
ranged as  follows: 


COMSTOCK  -  STUDIES    IN    ASTRONOMY  97 


sin  n  2  sin*  A  # 

a  =  —  -  x  =  —  :  —  -|—  cotip 

sin  p  sin  1" 


tan  M 


tan  (p  tan  i  (  p  —  x)  cos  M 
cos  N  =  . 


2  cosec  2  <p  <?o£  2V 


Since  cos  JV  =  cos  (  —  N)  the  algebraic  sign  of  N  is  not  de- 
termined by  the  equations,  but  it  is  apparent  from  the 
physical  conditions  of  the  problem  that  N  must  be  positive 
for  a  star  west  of  the  meridian  and  negative  for  a  star  east 
of  the  meridian.  The  coefficient  of  4  9  appears  in  equa- 
tions (13)  with  changed  sign  in  order  that  A  9  may  repre- 
sent a  correction  to  the  assumed  latitude.  In  the  use  of 
these  formulae  x  may  be  computed  with  four  place  logar- 
ithms, a  and  M  with  five  place,  and  N  with  six  or  seven 
place  tables. 

It  will  frequently  happen  that  the  observations  of  the 
polar  star  and  the  southern  star  composing  a  pair  will  be 
made  at  slightly  different  zenith  distances,  the  slight 
change  in  the  zenith  distance  of  the  line  of  sight  of  the 
telescope  being  indicated  by  an  altered  reading  of  the  level 
bubble.  This  alteration  is  most  conveniently  taken  into 
account  by  applying  to  the  observed  time,  Tz,  a  correction 

Level  Corr.  =  4^  z  =  4^f  •  ffil^iil  (U) 

dz  sin  A%  15 

where  &2  and  &1  are  the  level  readings,  r  the  value  of  a 
level  division  in  seconds  of  arc,  and  A2  the  azimuth  of  the 
star.  The  factor  &2  —  &:  is  to  be  considered  positive  when 
the  bubble  runs  from  its  initial  position  toward  the  objec- 
tive end  of  the  telescope. 

The  factor  sfc  ^   in  the  preceding  equation  may  be  re- 

S?>}1      *.*i  O 

placed  by  an  expression  which  is  most  conveniently  treated 
in  connection  with  the  thread  intervals. 


98  BULLETIN  OF  THE  UNIVERSITY  OF  WISCONSIN 

The  southern  star,  and  occasionally  the  polar  star,  will 
be  observed  on  several  threads,  and  from  the  several  ob- 
served times  the  time  of  transit  over  the  middle  thread 
may  be  found  by  Bessel's  method,1  or  as  follows:  The  re- 
duction of  any  thread  to  the  middle  thread  is  given  by  the 
equation 

' 


m  .  ,    tf  sec  &  cosec  Az  cot  A2 

=  T  +  i  sec  g>  cosec  A2  +  -»  --  :  -  —  -?  -  2  (15) 

2  sin  z  tan  q 

where  q  is  the  parallactic  angle  of  the  star  when  on  the 
middle  thread. 

When  the  star  is  observed  at  its  transit  over  the  almu- 
cantar  passing  through  the  pole,  we  have  rigorously 

q  =  t         z  =  90°  —  q> 

and  since  the  last  term  of  (15)  is  very  small  we  may  in 
most  cases  substitute  these  approximate  values  in  it. 
From  the  observations  on  the  first  and  last  threads  we  ob- 
tain, approximately, 

/  =  sec  q>  cosec  A%  =  (T"  —  T)  -t-  (i"  —  i')  (16) 

Applying  (15)  to  each  observed  thread  and  taking  the 
mean  of  the  resulting  equations,  we  obtain 


The  last  term  rarely  amounts  to  more  than  a  few  hun- 
dredths  of  a  second,  and  if  the  star  observed  is  near  the 
prime  vertical,  or  near  elongation,  it  may  be  neglected.  It 
should  be  noted  that  owing  to  the  factor  cosec  A2,  f  is  posi- 
tive for  stars  west  of  the  meridian  and  negative  for  stars 
east  of  the  meridian. 

Effect  of  Diurnal  Aberration.  —  The  effect  of  the  diurnal  ab- 
erration is  to  displace  every  star  toward  the  east  point  of 
the  horizon  by  the  amount 

s. 
D  =  0.021  cos  <p  sin  A 

i  Chauvenet,  Table  VIII. 


COMSTOCK  —  STUDIES    IN   ASTRONOMY  99 

where  A  is  the  angular  distance  of  the  star  from  the  east 
point.  If  in  the  quadrantal  triangle  formed  by  the  star, 
the  zenith  and  the  east  point  we  represent  the  angle  at  the 
star  by  ^  we  shall  have  for  the  effect  of  the  diurnal  aber- 
ration upon  the  time  of  the  star's  transit  over  an  almu- 
c  an  tar 

s. 
H  =  0.021  cos  (p  sin  A  .  cos  tp  .  sec  <p  cosec  A 


which  reduces  to 


s. 
=  0.021  cos  z 


or  for  an  observation  made  near  the  almucantar  passing 
through  the  pole 

s. 
K  =  0.021  sin  (p 

Since  the  effect  of  the  diurnal  aberration  is  thus  shown 
to  be  constant  for  all  stars  observed  at  the  same  zenith 
distance,  it  will  be  most  readily  taken  into  account  by  ap- 
plying to  the  clock  correction  derived  from  the  uncor- 
rected  observations  the  correction  +«. 

The  application  of  the  preceding  formulae  may  be  illus- 
trated by  the  reduction  of  the  following  observations  of 
four  pairs  of  stars,  made  with  a  very  small  universal  in- 
strument mounted  upon  a  portable  wooden  tripod.  The 
aperture  of  the  telescope  was  33  mm,  the  magnifying  power 
27  diameters,  the  value  of  a  level  division  7".  4: 


100 


BULLETIN  OF  THE  UNIVERSITY  OF  WISCONSIN 


WASHBURN  OBSERVATORY,   MAY  19,   1894. 
COMPABISON    OF     CLOCK     STABS     WITH    A.    UBS^    MINOBIS. 

Observer,  G.  C.  C. 


Star.  Circle. 

C  Leo.     L. 

49  Here.   L. 

f-L  Here.     R. 

3-  Leo.     R. 

&1 

6.0    24.7 

5.2    24.1 

5.0    23.9 

4.7    23.6 

T, 

13    31    5 

13    45    18 

13    56    4 

14    4    22 

^ 

5.1    24.0 

5.7    24.7 

5.0    24.1 

4.7    23.8 

T* 

13    40    43.24 

13    50    4.56 

14    0    1.43 

14    8    19.72 

0                  / 

O             ' 

0              1 

0          ' 

Azimuth  Circle 

83    35 

292    54 

270    18 

69    3 

f 

1.39 

1.51 

1.39 

1.50 

NOTES. —  The  instrument  was  so  set  that  the  readings  of 
the  azimuth  circle  are  very  nearly  the  true  azimuths  of  the 
line  of  sight.  When  the  vertical  circle  is  "Left "  the  zero 
of  the  level  scale  is  adjacent  to  the  objective.  The  symbol 
T2'  denotes  the  mean  of  the  observed  times  on  five  threads, 
and  requires  a  correction  to  reduce  it  to  T,.  T^  was  ob- 
served on  the  middle  thread  only.  The  thread  intervals 
are  assumed  to  be 

I  =  -  31.5  II  =  -  20.5  III  =  0.0  IV  =  4-  20.3  F  =  4-  30*8 
the  signs  corresponding  to  an  observation  Circle  R.  Star  W. 
It  should  be  noted  that  owing  to  the  reversal  of  the  instru- 
ment the  effect  of  any  small  error  in  the  adopted  thread 
intervals  will  be  eliminated. 

The  coordinates  of  A  Ursce  Hinoris  and  the  other  constants 
requisite  for  the  reduction,  are : 

ax   =19    29    25.6  log  sin  n   =  8.25491 

,      2  sin"  -£  it 


1 
43 


1    49.88 

5    (assumed) 


=1.5232 

sin  1 

Iog'2cosec2  g>  =  0.3020 


Hourly  rate  of  chronometer  =   —  0.04 


COMSTOCK STUDIES    IN    ASTRONOMY 


101 


Star. 

C  Leo. 

49  Here. 

ju  Here. 

S-  Leo. 

Retfn  to  Middle  Thread 

s.          s. 
+0.25  +0.00 

s.           s. 
-0.27  -0.02 

s.           s. 
+0.25  +0.00 

s.           s. 
-0.27  +0.02 

Chron.  Rate.    Level 

4-0.01  +0.54 

+0.01  +0.40 

0.00  -0.07 

-0.01  +0.07 

T* 

13  40  44.04 

13  50    4.68 

14    0    1.65 

14    8  19.53 

az 

10  10  49.05 

16  47  17.55 

17  42  20.71 

11    8  42.21 

«*  -  T& 

-3  29  54.99 

+2  57  12.87 

+3  42  19.06 

-2  59  37.32 

<*i  ~  TI 

+5  58  20.6 

+5  44    7.6 

+5  33  21  6 

+5  25    3.6 

t 

217  56    6 

318  16  20 

332  14  22 

233  49  47 

P 

66    3  13.9 

74  51    7.6 

62  13  18.7 

73  59  31.1 

log  sinp 

9.96091 

9.98464 

9.94682 

9.98283 

log  cos  t 

9.  89692  n 

9.87292 

9.94689 

9.  77099  n 

log  a 

8.29400 

8.27027 

8.30809 

8.27208 

log  sin  t 

9.  78871  n 

9.  82321  n 

9.  66818  n 

9.90702n 

co  log  (1  —  a  cost) 

9.9933111 

0.0060815 

0.0078831 

9.9952305 

log  cos  M 

9.9999692 

9.9999657 

9.9999798 

9.9999516 

H 

-0  40  57.1 

-0  43  13.9 

-0  33    8.7 

-0  51  21.2 

X 

51.3 

43.6 

55.3 

44.3 

log  tan  £  (p  —  x) 

9.8128460 

9.8837244 

9.7805448 

9.8769541 

\  cos  M  -T-  (1  —  a  cos  t)  I 

9.9932803 

0.0060472 

0.0078629 

9.9951821 

log  cos  N 

9.7770484 

9.8606937 

9.7593298 

9.8430583 

N 

+53  14  20.4 

-43  28  54.2 

-54  55  54.1 

+45  50    8.0 

M+N 

+  3  30  13.55 

-  2  56  48.54 

-  3  41  56.19 

+  2  59  55.12 

log  cot  N 

9.8733 

0.0230n 

9.8463n 

9.9873 

C 

+1.497 

-2.113 

-1.407 

+1.946 

s. 

s. 

s. 

s. 

AT+C  A  cp 

+18.56 

+24.33 

+22.87 

+17.80 

CAq> 

-  2.32 

+  3.27 

+  2.18 

-  3.02 

AT 

+20.88 

+21.06 

+20.69 

+20.82 

102  BULLETIN  OF  THE  UNIVERSITY  OF  WISCONSIN 

From  the  equations 

AT+  1M  Aq>  =  18.56  v  =  +0.02 

AT-Z.\\A(f>  =  24.33  +    .20 

AT-  1.41  Aq>  =  22.87  -     .17 

jr+1. 95  ^<p  =  17.80  -     .04 

we  obtain 

».  s. 
JT  =  +  20.86                  2f(?  =  -  1.55  =   -  23.3 

with  the  residuals  placed  opposite  the  several  equations. 

In  the  above  reduction  the  computations  have  been  car- 
ried to  tenths  of  a  second  of  arc  and  hundredths  of  a 
second  of  time,  but  it  is  evident  that  quantities  of  this  order 
are  imperceptible  in  so  small  an  instrument,  a  second  of 
arc  being  approximately  the  limit  of  what  can  be  seen  in 
its  telescope  or  measured  by  its  level.  The  internal  agree- 
ment of  the  observations  as  shown  by  the  residuals  is, 
therefore,  satisfactory,  and  the  absolute  values  of  the  lat- 
itude and  clock  correction  furnished  by  the  observations 
are  also  in  excellent  agreement  with  the  data  furnished  by 
a  geodetic  connection  with  the  "VVashburn  Observatory  and 
a  comparison  of  the  chronometer  with  the  normal  clock. 
Thus  after  correcting  AT  for  diurnal  aberration  we  have 

s.  °    ' 

From  Observation        AT  =  +  20.88  q>  =  43    4    36.7 

From  Comparisons  =  +20.80  =  43    4    36.5 

This  excellent  agreement  is  due,  at  least  in  part,  to  the 
reversal  of  the  instrument,  one-half  of  the  observations 
having  been  made  Circle  Right  and  one-half  Circle  Left, 
thus  eliminating  the  effect  of  error  in  the  assumed  thread 
intervals. 

In  order  to  secure  the  convenient  observation  of  stars  it 
will  be  advantageous  to  prepare  in  advance  an  observing 
programme  showing  the  time  at  which  the  several  clock 
stars  cross  the  almucantar  of  the  polar  star,  and  their  cor- 
responding azimuths.  If  only  a  few  stars  are  to  be  in- 
cluded in  the  programme  this  can  be  most  conveniently 
done  by  putting  T2=  T^  in  equations  (13)  and  solving  with 


COMSTOCK — STUDIES    IN    ASTRONOMY  103 

four  place  logarithms  the  following  approximate  equiva- 
lents of  those  equations : 

a  sin  t 


t  =  or 2  —  a±  tan  M  = 

a  =  sin  TT  cosec  j)  cos  N  = 


1  —  a  cos  £ 
tan  CD  tan  4 


\-acost 
T!    =    T2   ==   a2  -  AT  +  M  +  N 

When  the  sidereal  times  ^  and  T2  are  known  the  zenith 
distances  and  azimuths  of  the  stars  may  be  directly  com- 
puted from  the  fundamental  formula  for  the  transforma- 
tion of  coordinates,  but  the  following  method  will  usually 
be  found  more  convenient: 

In  the  spherical  triang]e  formed  by  the  polar  star,  the 
zenith  and  the  pole,  we  represent  the  east  hour  angle  of  the 
star  by  r  and  find 

cos  z  =  sin  cp  sin  d^  -f-  cos  (p  cos  d^  cos  T 

=  cos  (d1  —  <p)  —  cos  (p  cos  Si  2  sin2  $  r 
and  applying  to  this  the  development  into  series  of 

cos  x  =  cos  y  +  h 
find  when  terms  of  the  order  -H*  are  neglected 

z  =  H  -  <p  H  =   90°  -  7t  cos  T  (19) 

Similarly  from  the  development  of  the  azimuth  into 
series  we  find  when  the  azimuth  is  reckoned  from  the 
north,  positive  toward  east, 

AI  =  TC  sin  r  sec  (p  =  Jf0  sec  g>  (20) 

Values  of  H  and  If0  "with  the  argument  'r  are  tabulated 
below. 

To  determine  the  difference  of  azimuth  of  the  stars,  we 
represent  by  p  the  length  of  an  arc  of  a  great  circle  join- 
ing them,  and  from  the  isosceles  spherical  triangle  formed 
by  the  two  stars  and  the  zenith,  find 

cos  p  =  cos-z  -f-  sin?z  cos  (A2  —  AJ 
which  is  readily  transposed  into  either 

sin  \  (A2  —  A±)  =3  sin  |-  p  cosec  z 
or 

sin  i  p  (21) 


tan 


*\sin  (z  -  |)  sin  (z  + 


104  BULLETIN  OF  THE  UNIVERSITY  OF  WISCONSIN 

The  first  of  these  equations  will  usually  be  the  more  con- 
venient. 

To  determine  P  we  have  from  the  triangle  formed  by  the 
two  stars  and  the  pole 

cos  p  =  cos  it  sin  <$  -}-  sin  it  cos  S  cos  («2  —  a^) 
where  8  is  the  declination  of  the  southern  star.     In  place 
of  this  rigorous  equation  we  may  write  with  sufficient  pre- 
cision 

p  =  90  -  8  -  Tt  cos  (o-2  -  ax)  =  H (t)  -  d  (22) 

where  the  symbol  E  (t)  denotes  the  tabular  value  of  H  cor- 
responding to  the  argument  t  =  #2  —  <*i  • 

Equations  (18),  (20)  and  (22),  in  connection  with  the  tab- 
ular values  of  E  and  Jf0,  suffice  for  the  construction  of  an 
observing  list,  but  if  any  considerable  number  of  stars  are 
to  be  observed  in  the  same  latitude  it  will  be  found  an 
economy  of  labor  to  construct  for  the  given  latitude  special 
tables,  such  as  those  given  below  for  the  Washfturn  Ob- 
servatory, which  are  based  on  the  following  analysis : 

Neglecting  terms  of  the  order  n-  we  put 

cos  N0   =  tan  cp  tan  |  p 
and  find  from  equations  (18) 

cos  N  =  cos  NQ  +  a  cos  t  cos  N0 
N  =  NQ  —  Tt  cos  t  sec  S  cot  N0 
M  =  it  sin  t  sec  d 

The  factor  *  sin  t  =  J/0  has  been  tabulated,  and  —  *  cos  t  is 
evidently  equal  to  the  tabular  value  of  M0  which  corres- 
ponds to  the  argument  t  —  67i .  Putting 

sec  d  =  h  sec  8  cot  N0   =  k 

we  tabulate  N0,  h  and  k  with  the  argument  s  and  find  for 
the  instant  at  which  the  two  stars  have  equal  altitudes 

T  =  a  +  N0  +  h  J/'o  +  k  M\ 

the  accents  '  "  denoting  that  the  arguments  for  the  cor- 
responding values  of  M0  are  t  and  t  —  Qh . 

It  should  be  noted  that  since  N0  is  an  approximation  to 
the  N  of  the  rigorous  formulae  we  shall  have  N0  and  k  posi- 


COMSTOCK — STUDIES    IN    ASTRONOMY 


105 


tive  for  a  star  west  of  the  meridian  and  negative  for  a  star 
east  of  the  meridian. 

Similar  tables  may  be  constructed  for  the  difference  of 
azimuth  of  the  stars,  but  the  direct  computation  by  (19) 
and  (21)  is  so  simple  that  little  advantage  would  be  derived 
from  them. 

To  illustrate  the  use  of  the  tables  we  make  the  following 
computations  for  a  comparison  of  Polaris  with  p  Leonis 
west  of  the  meridian  and  d  Herculis  east  of  the  meridian: 


<P  —  43    4.6 


h,     m. 

1    18.9 


Star, 

p  Leonis. 

d  Herculis. 

a2 

10    27.3 

17    10.7 

t 

9      8.4 

15    51.8 

d 

9    51 

24    58 

H(t) 

90    54 

90    39 

P 

81      3 

65    41 

X0 

-f  2    32.5 

-  3    33.7 

hM0' 

H-         3.4 

4.6 

kM0" 

+         4.6 

2.1 

T 

13      7.8 

13    30.3 

H(T) 

91    14 

91    14 

z 

48    10 

48    10 

cosec  z 

0.1278 

0.1278 

sinp 

9.8127 

9.7343 

2 

A,-  A, 

121    23 

93    25 

The  azimuth  of  Polaris  corresponding  to  the  times  above 
computed  is,  in  minutes  of  arc,  15  sec  9-  MJ. 

For  a  determination  of  time  only,  the  latitude  being  sup- 
posed known  or  not  required,  the  observation  of  the  polar 


106 


BULLETIN  OF  THE  UNIVERSITY  OF  WISCONSIN 


star  may  be  omitted  and  the  observation  confined  to  noting 
the  times  at  which  two  southern  stars  reach  the  same  alti- 
tude. Convenient  formulae  and  tables  for  observations  of 
this  kind  have  been  published  by  Wittram. l 

AUXILIARY  TABLES  FOR  TRANSITS  OVER  THE  ALMU- 
CANTAR  OF  POLARIS. 

FOB  ALL  LATITUDES. 


t  or  T 

M0 

H 

t  or  T 

h. 

m. 

O     ' 

h. 

0 

+  0.0  - 

88  46 

24 

1.3 

3 

1 

+  1.3  - 

88  49 

23 

1.2 

7 

2 

+  2.5  - 

88  56 

22 

1.0 

12 

3 

+  3.5  - 

89   8 

21 

0.8 

15 

4 

+  4.3  - 

89  23 

20 

0.5 

18 

5 

+  4.8  - 

89  41 

19 

0.1 

19 

6 

+  4.9  - 

90   0 

18 

0.1 

19 

7 

+  4.8  - 

90  19 

17 

0.5 

18 

8 

+  4.3  - 

90  37 

16 

0.8 

15 

9 

+  3.5  - 

90  52 

15 

1.0 

12 

10 

+  2.5  - 

91   4 

14 

1.2 

7 

11 

+  1.3  - 

91  11 

13 

1.3 

3 

12- 

+  00- 

91  14 

12 

i  Tables  Auxiliaires  pour  la  Determination  de  THeure  par  des  Hauteurs  Correspondantes 
de  Differentes  Etoiles.    St.  Petersburg,  1892. 


COMSTOCK — STUDIES    IN    ASTRONOMY 
FOB   THE   LATITUDE   43°   4: . '  6 


107 


s 

1 
ATo 

+  W.    -  E. 

h 

k 
+  W.  -  E. 

a 

0 

h.  m. 

• 

-  2 

0  57.9 

1.00 

3.88 

-  2 

13.8 

79 

-  1 

1  11.7 

1.00 

3.09 

-  1 

11.4 

45 

0 

1  23.1 

1.00 

2.64 

0 

9.9 

31 

+  1 

1  33.0 

1.00 

2.33 

+  1 

8.9 

23 

2 

1  41.9 

1.00 

2.10 

2 

8.0 

17 

3 

1  49.9 

1.00 

1.93 

3 

7.3 

15 

4 

1  57.2 

1.00 

1.78 

4 

6.9 

11 

5 

2   4.1 

1.00 

1.67 

5 

6.6 

10 

6 

2  10.7 

1.01 

1  57 

6 

6.1 

9 

7 

2  16  8 

1.01 

1.48 

7 

5.7 

7 

8 

2  22  5 

1.01 

1.41 

8 

5.5 

7 

9 

2  28.0 

1.01 

1.34 

9 

5.3 

6 

*  10  * 

2  33.3     *  * 

*  1.02  * 

1.28   *  * 

*  10  * 

9.8 

10 

12 

2  43.1 

1.02 

1.18 

12 

9.2 

8 

14 

2  52.3 

1.03 

1.10 

14 

8.5 

7 

16 

3   0.8 

1.04 

1.03 

16 

8.0 

6 

18 

3   8.8 

1.05 

0.97 

18 

7.6 

5 

20 

3  16.4 

1.06 

0.92 

20 

7.2 

4 

22 

3  23.6 

1.08 

0.88 

22 

6.9 

4 

24 

3  30.5 

1.09 

0.81 

24 

6.5 

4 

26 

3  37.0 

1.11 

0.80 

26 

6.3 

3 

28 

3  43.3 

1.13 

0.77 

28 

6.0 

3 

30 

3  49.3 

1.15 

0.74 

30 

5.8 

3 

32 

3  55.1 

1.18 

0.71 

32 

5.7 

2 

34 

4   0.8 

1.21 

0.69 

34 

5.4 

2 

36 

4   6.2 

1.24 

0.67 

36 

5.3 

2 

38 

4  11.5 

1.27 

0.65 

38 

5.1 

2 

40 

4  16.6 

1.31      0.63 

40 

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